## [Maxima-commits] CVS: maxima/share/contrib/diffequations/tests rtestode_kamke_2_1.mac, 1.2, 1.3 rtestode_kamke_2_2.mac, 1.4, 1.5 rtestode_kamke_2_3.mac, 1.3, 1.4

 [Maxima-commits] CVS: maxima/share/contrib/diffequations/tests rtestode_kamke_2_1.mac, 1.2, 1.3 rtestode_kamke_2_2.mac, 1.4, 1.5 rtestode_kamke_2_3.mac, 1.3, 1.4 From: David Billinghurst - 2007-01-31 12:44:22 ```Update of /cvsroot/maxima/maxima/share/contrib/diffequations/tests In directory sc8-pr-cvs7.sourceforge.net:/tmp/cvs-serv18653 Modified Files: rtestode_kamke_2_1.mac rtestode_kamke_2_2.mac rtestode_kamke_2_3.mac Log Message: odelin solves many 2nd order linear ODEs. Update testsuite. Index: rtestode_kamke_2_1.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_2_1.mac,v retrieving revision 1.2 retrieving revision 1.3 diff -u -d -r1.2 -r1.3 --- rtestode_kamke_2_1.mac 2 Sep 2006 12:22:36 -0000 1.2 +++ rtestode_kamke_2_1.mac 31 Jan 2007 12:44:11 -0000 1.3 @@ -93,19 +93,29 @@ /* Kamke 2. 10 */ (pn_(10),ans:contrib_ode(eqn:'diff(y,x,2)+(a*x+b)*y,y,x)); -false; +[y = bessel_y(1/3,2*(a*x+b)^(3/2)/(3*a))*%k2*sqrt(a*x+b) + +bessel_j(1/3,2*(a*x+b)^(3/2)/(3*a))*%k1*sqrt(a*x+b)]; +method; +odelin; /* Kamke 2. 11 */ (pn_(11),ans:contrib_ode(eqn:'diff(y,x,2)-(x^2+1)*y,y,x)); -false; +[y = %k2*%e^(x^2/2)*erf(x)+%k1*%e^(x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 12 */ (pn_(12),ans:contrib_ode(eqn:'diff(y,x,2)-(x^2+a)*y,y,x)); -false; +[y = %k2*kummer_u(-(a-1)/4,1/2,-x^2)*%e^(x^2/2) + +%k1*kummer_m(-(a-1)/4,1/2,-x^2)*%e^(x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 13 */ (pn_(13),ans:contrib_ode(eqn:'diff(y,x,2)-(a^2*x^2+a)*y,y,x)); -false; +[y = %k2*%e^(a*x^2/2)*erf(sqrt(a)*x)+%k1*%e^(a*x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 14 */ (pn_(14),ans:contrib_ode(eqn:'diff(y,x,2)-c*x^a*y,y,x)); @@ -225,7 +235,10 @@ /* Kamke 2. 37 */ (pn_(37),ans:contrib_ode(eqn:'diff(y,x,2)+a*'diff(y,x,1)+(-b^2*x^2-c)*y,y,x)); -false; +[y = %k2*kummer_u(-(4*c-4*b+a^2)/(16*b),1/2,-b*x^2)*%e^(x*(b*x-a)/2) + +%k1*kummer_m(-(4*c-4*b+a^2)/(16*b),1/2,-b*x^2)*%e^(x*(b*x-a)/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 38 */ (pn_(38),ans:contrib_ode(eqn:'diff(y,x,2)+2*a*'diff(y,x,1)+f(x)*y,y,x)); @@ -241,39 +254,61 @@ /* Kamke 2. 40 */ (pn_(40),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)-y,y,x)); -false; +[y = kummer_u(0,3/2,-x^2/2)*%k2*x+kummer_m(0,3/2,-x^2/2)*%k1*x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 41 */ (pn_(41),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)+(n+1)*y,y,x)); -false; +[y = %k2*kummer_u((n+1)/2,1/2,-x^2/2)+%k1*kummer_m((n+1)/2,1/2,-x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 42 */ (pn_(42),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)-n*y,y,x)); -false; +[y = %k2*kummer_u(-n/2,1/2,-x^2/2)+%k1*kummer_m(-n/2,1/2,-x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 43 */ (pn_(43),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)+2*y,y,x)); -false; +[y = kummer_u(2,3/2,-x^2/2)*%k2*x*%e^(x^2/2) + +kummer_m(2,3/2,-x^2/2)*%k1*x*%e^(x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 44 */ (pn_(44),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)-a*y,y,x)); -false; +[y = %k2*kummer_u(-(a-1)/2,1/2,-x^2/2)*%e^(x^2/2) + +%k1*kummer_m(-(a-1)/2,1/2,-x^2/2)*%e^(x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 45 */ (pn_(45),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)+(x-1)*y,y,x)); -false; +[y = %k2*%e^x+%k1*erf(%i*(x-2)/sqrt(2))*%e^(x-2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 46 */ (pn_(46),ans:contrib_ode(eqn:'diff(y,x,2)-2*x*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*kummer_u((a+2)/4,1/2,-x^2)*%e^x^2 + +%k1*kummer_m((a+2)/4,1/2,-x^2)*%e^x^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 47 */ (pn_(47),ans:contrib_ode(eqn:'diff(y,x,2)+4*x*'diff(y,x,1)+(4*x^2+2)*y,y,x)); -false; +[y = %k2*x*%e^-x^2+%k1*%e^-x^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 48 */ (pn_(48),ans:contrib_ode(eqn:'diff(y,x,2)-4*x*'diff(y,x,1)+(3*x^2+2*n-1)*y,y,x)); -false; +[y = %k2*kummer_u((n+1)/2,1/2,-x^2)*%e^(3*x^2/2) + +%k1*kummer_m((n+1)/2,1/2,-x^2)*%e^(3*x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 49 */ (pn_(49),ans:contrib_ode(eqn:'diff(y,x,2)-4*x*'diff(y,x,1)+(4*x^2-1)*y-%e^x,y,x)); @@ -281,7 +316,9 @@ /* Kamke 2. 50 */ (pn_(50),ans:contrib_ode(eqn:'diff(y,x,2)-4*x*'diff(y,x,1)+(4*x^2-2)*y,y,x)); -false; +[y = %k2*x*%e^x^2+%k1*%e^x^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 51 */ (pn_(51),ans:contrib_ode(eqn:'diff(y,x,2)-4*x*'diff(y,x,1)+(4*x^2-3)*y-%e^x^2,y,x)); @@ -289,39 +326,60 @@ /* Kamke 2. 52 */ (pn_(52),ans:contrib_ode(eqn:'diff(y,x,2)+a*x*'diff(y,x,1)+b*y,y,x)); -false; + [y = %k2*kummer_u(b/(2*a),1/2,-a*x^2/2) + +%k1*kummer_m(b/(2*a),1/2,-a*x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 53 */ (pn_(53),ans:contrib_ode(eqn:'diff(y,x,2)+2*a*x*'diff(y,x,1)+a^2*x^2*y,y,x)); -false; +[y = %k2*%e^-(a*x^2/2+sqrt(a)*x)+%k1*%e^(x*(2*sqrt(a)-a*x)/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 54 */ +/* CHECKME: Takes a long time (pn_(54),ans:contrib_ode(eqn:'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+(c*x+d)*y,y,x)); false; +*/ /* Kamke 2. 55 */ +/* CHECKME: Takes a long time (pn_(55),ans:contrib_ode(eqn:'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+(a1*x^2+b1*x+c_1)*y,y,x)); false; +*/ /* Kamke 2. 56 */ (pn_(56),ans:contrib_ode(eqn:'diff(y,x,2)-x^2*'diff(y,x,1)+x*y,y,x)); -false; +[y = kummer_u(1,2/3,-x^3/3)*%k2*%e^(x^3/3) + +kummer_m(1,2/3,-x^3/3)*%k1*%e^(x^3/3)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 57 */ (pn_(57),ans:contrib_ode(eqn:'diff(y,x,2)-x^2*'diff(y,x,1)-(x+1)^2*y,y,x)); -false; +[y = %k2*%e^(x*(x^2+3)/3)*'integrate(%e^(-x^3/3-2*x),x) + +%k1*%e^(x*(x^2+3)/3)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 58 */ (pn_(58),ans:contrib_ode(eqn:'diff(y,x,2)-x^2*(x+1)*'diff(y,x,1)+x*(x^4-2)*y,y,x)); -false; +[y = %k2*%e^(x^3/3)*'integrate(%e^(x^4/4-x^3/3),x)+%k1*%e^(x^3/3)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 59 */ (pn_(59),ans:contrib_ode(eqn:'diff(y,x,2)+x^4*'diff(y,x,1)-x^3*y,y,x)); -false; +[y = kummer_u(0,6/5,-x^5/5)*%k2*x+kummer_m(0,6/5,-x^5/5)*%k1*x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; -/* Kamke 2. 60 */ +/* Kamke 2. 60 */ +/* CHECKME: Takes a long time (pn_(60),ans:contrib_ode(eqn:'diff(y,x,2)+a*x^(q-1)*'diff(y,x,1)+b*x^(q-2)*y,y,x)); false; +*/ /* Kamke 2. 61 */ (pn_(61),ans:contrib_ode(eqn:'diff(y,x,2)+sqrt(x)*'diff(y,x,1)+(x/4+1/(4*sqrt(x))-9)*y-x*%e^-(x^(3/2)/3),y,x)); @@ -441,11 +499,16 @@ /* Kamke 2. 86 */ (pn_(86),ans:contrib_ode(eqn:4*'diff(y,x,2)+9*x*y,y,x)); -false; +[y = bessel_y(1/3,x^(3/2))*%k2*sqrt(x)+bessel_j(1/3,x^(3/2))*%k1*sqrt(x)]; +method; +odelin; /* Kamke 2. 87 */ (pn_(87),ans:contrib_ode(eqn:4*'diff(y,x,2)-(x^2+a)*y,y,x)); -false; +[y = %k2*kummer_u(-(a-2)/8,1/2,-x^2/2)*%e^(x^2/4) + +%k1*kummer_m(-(a-2)/8,1/2,-x^2/2)*%e^(x^2/4)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 88 */ (pn_(88),ans:contrib_ode(eqn:4*'diff(y,x,2)+4*tan(x)*'diff(y,x,1)+(-5*tan(x)^2-2)*y,y,x)); @@ -453,7 +516,12 @@ /* Kamke 2. 89 */ (pn_(89),ans:contrib_ode(eqn:a*'diff(y,x,2)-(x+c+a*b)*'diff(y,x,1)+(b*(x+c)+d)*y,y,x)); -false; +[y=%k2*kummer_u((d+1)/2,1/2,-(x^2+(2*c-2*a*b)*x+c^2-2*a*b*c+a^2*b^2)/(2*a)) + *%e^((2*x^2+4*c*x+c^2-2*a*b*c+a^2*b^2)/(4*a)) + +%k1*kummer_m((d+1)/2,1/2,-(x^2+(2*c-2*a*b)*x+c^2-2*a*b*c+a^2*b^2)/(2*a)) + *%e^((2*x^2+4*c*x+c^2-2*a*b*c+a^2*b^2)/(4*a))]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 90 */ (pn_(90),ans:contrib_ode(eqn:a^2*'diff(y,x,2)+a*(a^2-2*b*%e^-(a*x))*'diff(y,x,1)+b^2*%e^-(2*a*x)*y,y,x)); @@ -475,7 +543,10 @@ /* Kamke 2. 92 */ (pn_(92),ans:contrib_ode(eqn:x*'diff(y,x,2)+(x+a)*y,y,x)); -false; +[y = %k2*kummer_u(-%i*a/2,0,-2*%i*x)*%e^(%i*x) + +%k1*kummer_m(-%i*a/2,0,-2*%i*x)*%e^(%i*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 93 */ (pn_(93),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1),y,x)); @@ -487,19 +558,30 @@ /* Kamke 2. 94 */ (pn_(94),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1)+a*y,y,x)); -false; +[y = bessel_y(0,2*sqrt(a)*sqrt(x))*%k2 + +bessel_j(0,2*sqrt(a)*sqrt(x))*%k1]; +method; +odelin; /* Kamke 2. 95 */ (pn_(95),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1)+l*x*y,y,x)); -false; +[y = bessel_y(0,-sqrt(l)*x)*%k2+bessel_j(0,-sqrt(l)*x)*%k1]; +method; +odelin; /* Kamke 2. 96 */ (pn_(96),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1)+(x+a)*y,y,x)); -false; +[y = %k2*kummer_u(-(%i*a-1)/2,1,-2*%i*x)*%e^(%i*x) + +%k1*kummer_m(-(%i*a-1)/2,1,-2*%i*x)*%e^(%i*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 97 */ (pn_(97),ans:contrib_ode(eqn:x*'diff(y,x,2)-'diff(y,x,1)+a*y,y,x)); -false; +[y = bessel_y(-2,2*sqrt(a)*sqrt(x))*%k2*x + +bessel_j(-2,2*sqrt(a)*sqrt(x))*%k1*x]; +method; +odelin; /* Kamke 2. 98 */ assume(a>0); Index: rtestode_kamke_2_2.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_2_2.mac,v retrieving revision 1.4 retrieving revision 1.5 diff -u -d -r1.4 -r1.5 --- rtestode_kamke_2_2.mac 4 Sep 2006 12:27:16 -0000 1.4 +++ rtestode_kamke_2_2.mac 31 Jan 2007 12:44:11 -0000 1.5 @@ -11,23 +11,37 @@ /* Kamke 2. 101 */ (pn_(101),ans:contrib_ode(eqn:x*'diff(y,x,2)+2*'diff(y,x,1)+a*x*y,y,x)); -false; +[y = %k2*%e^(sqrt(-a)*x)/x+%k1*%e^-(sqrt(-a)*x)/x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 102 */ (pn_(102),ans:contrib_ode(eqn:x*'diff(y,x,2)+2*'diff(y,x,1)+a*x^2*y,y,x)); -false; +[y = bessel_y(1/3,2*sqrt(a)*x^(3/2)/3)*%k2/sqrt(x) + +bessel_j(1/3,2*sqrt(a)*x^(3/2)/3)*%k1/sqrt(x)]; +method; +odelin; /* Kamke 2. 103 */ (pn_(103),ans:contrib_ode(eqn:x*'diff(y,x,2)-2*'diff(y,x,1)+a*y,y,x)); -false; +[y = bessel_y(-3,2*sqrt(a)*sqrt(x))*%k2*x^(3/2) + +bessel_j(-3,2*sqrt(a)*sqrt(x))*%k1*x^(3/2)]; +method; +odelin; /* Kamke 2. 104 */ (pn_(104),ans:contrib_ode(eqn:x*'diff(y,x,2)+v*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*bessel_y(v-1,2*sqrt(a)*sqrt(x))/x^(v/2-1/2) + +%k1*bessel_j(v-1,2*sqrt(a)*sqrt(x))/x^(v/2-1/2)]; +method; +odelin; /* Kamke 2. 105 */ (pn_(105),ans:contrib_ode(eqn:x*'diff(y,x,2)+a*'diff(y,x,1)+b*x*y,y,x)); -false; + [y = %k2*bessel_y(-(a-1)/2,-sqrt(b)*x)/x^(a/2-1/2) + +%k1*bessel_j(-(a-1)/2,-sqrt(b)*x)/x^(a/2-1/2)]; +method; +odelin; /* Kamke 2. 106 */ (pn_(106),ans:contrib_ode(eqn:x*'diff(y,x,2)+a*'diff(y,x,1)+b*x^a1*y,y,x)); @@ -35,11 +49,15 @@ /* Kamke 2. 107 */ (pn_(107),ans:contrib_ode(eqn:x*'diff(y,x,2)+(x+b)*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*kummer_u(a,b,-x)+%k1*kummer_m(a,b,-x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 108 */ (pn_(108),ans:contrib_ode(eqn:x*'diff(y,x,2)+(x+b+a)*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*kummer_u(a,b+a,-x)+%k1*kummer_m(a,b+a,-x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 109 */ (pn_(109),ans:contrib_ode(eqn:x*'diff(y,x,2)-x*'diff(y,x,1)-y-x*(x+1)*%e^x,y,x)); @@ -49,71 +67,97 @@ method; variationofparameters; -/* Kamke 2. 110 */ +/* Kamke 2. 110 */ assume(x>0,x+4*a>0); [x>0,x+4*a>0]; (pn_(110),ans:contrib_ode(eqn:x*'diff(y,x,2)-x*'diff(y,x,1)-a*y,y,x)); -[[]]; -method; -freeofx; +[y = %k2*kummer_u(-a,0,-x)*%e^x+%k1*kummer_m(-a,0,-x)*%e^x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; forget(x>0,x+4*a>0); [x>0,x+4*a>0]; -/* Kamke 2. 111 */ +/* Kamke 2. 111 */ (pn_(111),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x+1)*'diff(y,x,1)+y,y,x)); -[[]]; -method; -freeofx; +[y = %k2*%e^x+%k1*(x+1)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 112 */ -(pn_(112),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x+1)*'diff(y,x,1)-2*(x-1)*y,y,x)); -false; +(pn_(112),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x+1)*'diff(y,x,1)-2*(x-1)*y,y,x));\ +[y = %k2*%e^(2*x)+%k1*(3*x+1)*%e^-x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 113 */ assume(x^2+(4*a-2*b)*x+b^2>0); [x^2+(4*a-2*b)*x+b^2>0]; (pn_(113),ans:contrib_ode(eqn:x*'diff(y,x,2)+(b-x)*'diff(y,x,1)-a*y,y,x)); -[[]]; -method; -freeofx; +[y = %k2*kummer_u(b-a,b,-x)*%e^x+%k1*kummer_m(b-a,b,-x)*%e^x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; forget(x^2+(4*a-2*b)*x+b^2>0); [x^2+(4*a-2*b)*x+b^2>0]; /* Kamke 2. 114 */ (pn_(114),ans:contrib_ode(eqn:x*'diff(y,x,2)-2*(x-1)*'diff(y,x,1)-y,y,x)); -false; +[y = kummer_u(1/2,0,-2*x)*%k2*%e^(2*x)/x + +kummer_m(1/2,0,-2*x)*%k1*%e^(2*x)/x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 115 */ (pn_(115),ans:contrib_ode(eqn:x*'diff(y,x,2)-(3*x-2)*'diff(y,x,1)+(3-2*x)*y,y,x)); -false; +[y = kummer_u(6*sqrt(17)/17,0,-sqrt(17)*x)*%k2*%e^((sqrt(17)+3)*x/2)/x + +kummer_m(6*sqrt(17)/17,0,-sqrt(17)*x)*%k1*%e^((sqrt(17)+3)*x/2)/x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 116 */ -(pn_(116),ans:contrib_ode(eqn:x*'diff(y,x,2)+(a*x+n+b)*'diff(y,x,1)+a*n*y,y,x)); -false; +(pn_(116),ans:contrib_ode(eqn:x*'diff(y,x,2)+(a*x+n+b)*'diff(y,x,1)+a*n*y,y,x));[y = %k2*kummer_u(n,n+b,-a*x)+%k1*kummer_m(n,n+b,-a*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 117 */ (pn_(117),ans:contrib_ode(eqn:x*'diff(y,x,2)-(b+a)*(x+1)*'diff(y,x,1)+a*b*x*y,y,x)); -false; +[y = %k2*kummer_u(-b*(b+a)/(b-a),-b-a,-(b-a)*x)*%e^(b*x) + +%k1*kummer_m(-b*(b+a)/(b-a),-b-a,-(b-a)*x)*%e^(b*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 118 */ (pn_(118),ans:contrib_ode(eqn:x*'diff(y,x,2)+((b+a)*x+n+m)*'diff(y,x,1)+(a*b*x+a*n+b*m)*y,y,x)); -false; +[y = %k2*kummer_u(m,n+m,-(b-a)*x)*%e^-(a*x) + +%k1*kummer_m(m,n+m,-(b-a)*x)*%e^-(a*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; -/* Kamke 2. 119 */ +/* Kamke 2. 119 */ +/* CHECKME: Takes a long time (pn_(119),ans:contrib_ode(eqn:x*'diff(y,x,2)-2*(a*x+b)*'diff(y,x,1)+(a^2*x+2*a*b)*y,y,x)); false; +*/ /* Kamke 2. 120 */ (pn_(120),ans:contrib_ode(eqn:x*'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+(c*x+d)*y,y,x)); -false; +[y = %k2*kummer_u(-(sqrt(a^2-4*c)*(2*d-a*b)-4*b*c+a^2*b)/(2*(4*c-a^2)), + b,-sqrt(a^2-4*c)*x)*%e^((sqrt(a^2-4*c)-a)*x/2) + +%k1*kummer_m(-(sqrt(a^2-4*c)*(2*d-a*b)-4*b*c+a^2*b)/(2*(4*c-a^2)), + b,-sqrt(a^2-4*c)*x)*%e^((sqrt(a^2-4*c)-a)*x/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 121 */ (pn_(121),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x^2-x)*'diff(y,x,1)+(x-1)*y,y,x)); -false; +[y = %k2*x*'integrate(%e^(x^2/2-x)/x^2,x)+%k1*x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 122 */ (pn_(122),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x^2-x-2)*'diff(y,x,1)-x*(x+3)*y,y,x)); -false; +[y = %k2*%e^(x^2/2)*'integrate(%e^(-x^2/2-x)/x^2,x)+%k1*%e^(x^2/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 123 */ assume(b>0,b>a^2); @@ -129,7 +173,9 @@ /* Kamke 2. 124 */ (pn_(124),ans:contrib_ode(eqn:x*'diff(y,x,2)-2*(x^2-a)*'diff(y,x,1)+2*n*x*y,y,x)); -false; +[y = %k2*kummer_u(-n/2,(2*a+1)/2,x^2)+%k1*kummer_m(-n/2,(2*a+1)/2,x^2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 125 */ (pn_(125),ans:contrib_ode(eqn:x*'diff(y,x,2)+(4*x^2-1)*'diff(y,x,1)-4*x^3*y-4*x^5,y,x)); @@ -142,8 +188,10 @@ variationofparameters; /* Kamke 2. 126 */ +/* CHECKME: Takes a long time (pn_(126),ans:contrib_ode(eqn:x*'diff(y,x,2)+(2*a*x^3-1)*'diff(y,x,1)+(a^2*x^3+a)*y,y,x)); false; +*/ /* Kamke 2. 127 */ (pn_(127),ans:contrib_ode(eqn:x*'diff(y,x,2)+(2*a*x*log(x)+1)*'diff(y,x,1)+(a^2*x*log(x)^2+a*log(x)+a)*y,y,x)); @@ -155,7 +203,9 @@ /* Kamke 2. 129 */ (pn_(129),ans:contrib_ode(eqn:(x-3)*'diff(y,x,2)-(4*x-9)*'diff(y,x,1)+(3*x-6)*y,y,x)); -false; +[y = %k2*(4*x^3-42*x^2+150*x-183)*%e^(3*x)+%k1*%e^x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 130 */ (pn_(130),ans:contrib_ode(eqn:2*x*'diff(y,x,2)+'diff(y,x,1)+a*y,y,x)); @@ -167,25 +217,33 @@ /* Kamke 2. 131 */ (pn_(131),ans:contrib_ode(eqn:2*x*'diff(y,x,2)-(x-1)*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*kummer_u(a+1,3/2,-x/2)*sqrt(x)*%e^(x/2) + +%k1*kummer_m(a+1,3/2,-x/2)*sqrt(x)*%e^(x/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 132 */ assume(4*x^2+(-8*a-4)*x+1>0); [4*x^2+(-8*a-4)*x+1>0]; (pn_(132),ans:contrib_ode(eqn:2*x*'diff(y,x,2)-(2*x-1)*'diff(y,x,1)+a*y,y,x)); -[[]]; -method; -freeofx; +[y = %k2*kummer_u((a+2)/2,3/2,-x)*sqrt(x)*%e^x + +%k1*kummer_m((a+2)/2,3/2,-x)*sqrt(x)*%e^x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; forget(4*x^2+(-8*a-4)*x+1>0); [4*x^2+(-8*a-4)*x+1>0]; /* Kamke 2. 133 */ (pn_(133),ans:contrib_ode(eqn:(2*x-1)*'diff(y,x,2)-(3*x-4)*'diff(y,x,1)+(x-3)*y,y,x)); -false; +[y = %k2*%e^x*'integrate(%e^-(x/2)/(2*x-1)^(5/4),x)+%k1*%e^x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 134 */ (pn_(134),ans:contrib_ode(eqn:4*x*'diff(y,x,2)-(x+a)*y,y,x)); -false; +[y = %k2*kummer_u(-a/4,0,-x)*%e^(x/2)+%k1*kummer_m(-a/4,0,-x)*%e^(x/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 135 */ (pn_(135),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+2*'diff(y,x,1)-y,y,x)); @@ -197,27 +255,40 @@ /* Kamke 2. 136 */ (pn_(136),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+4*'diff(y,x,1)+(-x-2)*y,y,x)); -false; +[y = %k2*%e^(x/2)*'integrate(%e^-x/x,x)+%k1*%e^(x/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 137 */ (pn_(137),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+(-x-2)*y+l*y+4*y,y,x)); -false; +[y = %k2*kummer_u((l+2)/4,0,-x)*%e^(x/2) + +%k1*kummer_m((l+2)/4,0,-x)*%e^(x/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 138 */ (pn_(138),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+4*m*'diff(y,x,1)+(-x+4*n+2*m)*y,y,x)); -false; +[y = %k2*kummer_u(n+m,m,-x)*%e^(x/2)+%k1*kummer_m(n+m,m,-x)*%e^(x/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 139 */ (pn_(139),ans:contrib_ode(eqn:16*x*'diff(y,x,2)+8*'diff(y,x,1)+(-x-a)*y,y,x)); -false; +[y = %k2*kummer_u(-(a-6)/8,3/2,-x/2)*sqrt(x)*%e^(x/4) + +%k1*kummer_m(-(a-6)/8,3/2,-x/2)*sqrt(x)*%e^(x/4)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 140 */ (pn_(140),ans:contrib_ode(eqn:a*x*'diff(y,x,2)+b*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*bessel_y((b-a)/a,2*sqrt(c)*sqrt(x)/sqrt(a))/x^(b/(2*a)-1/2) + +%k1*bessel_j((b-a)/a,2*sqrt(c)*sqrt(x)/sqrt(a))/x^(b/(2*a)-1/2)]; /* Kamke 2. 141 */ (pn_(141),ans:contrib_ode(eqn:a*x*'diff(y,x,2)+(b*x+3*a)*'diff(y,x,1)+3*b*y,y,x)); -false; +[y = %k2*%e^-(b*x/a)*'integrate(%e^(b*x/a)/x^3,x)+%k1*%e^-(b*x/a)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 142 */ (pn_(142),ans:contrib_ode(eqn:5*(a*x+b)*'diff(y,x,2)+8*a*'diff(y,x,1)+c*(a*x+b)^(1/5)*y,y,x)); @@ -225,15 +296,32 @@ /* Kamke 2. 143 */ (pn_(143),ans:contrib_ode(eqn:2*a*x*'diff(y,x,2)+(b*x+a)*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*kummer_u(c/b,1/2,-b*x/(2*a))+%k1*kummer_m(c/b,1/2,-b*x/(2*a))]; +[method, ode_check(eqn,ans[1])]; +odelin,0]; /* Kamke 2. 144 */ (pn_(144),ans:contrib_ode(eqn:2*a*x*'diff(y,x,2)+(b*x+3*a)*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*kummer_u(c/b,3/2,-b*x/(2*a))+%k1*kummer_m(c/b,3/2,-b*x/(2*a))]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; -/* Kamke 2. 145 */ +/* Kamke 2. 145 */ (pn_(145),ans:contrib_ode(eqn:(a2*x+b2)*'diff(y,x,2)+(a1*x+b1)*'diff(y,x,1)+(a0*x+b0)*y,y,x)); -false; +[y = %k2*kummer_u((sqrt(a1^2-4*a0*a2) + *((2*a0*a2-a1^2)*b2+a1*a2*b1-2*a2^2*b0) + +(a1^3-4*a0*a1*a2)*b2+(4*a0*a2^2-a1^2*a2)*b1) + /(2*a2^2*(4*a0*a2-a1^2)),-(a1*b2-a2*b1)/a2^2, + -sqrt(a1^2-4*a0*a2)*(a2*x+b2)/a2^2) + *%e^((sqrt(a1^2-4*a0*a2)*(a2*x+b2)-a1*a2*x)/(2*a2^2)) + +%k1*kummer_m((sqrt(a1^2-4*a0*a2) + *((2*a0*a2-a1^2)*b2+a1*a2*b1-2*a2^2*b0) + +(a1^3-4*a0*a1*a2)*b2+(4*a0*a2^2-a1^2*a2)*b1) + /(2*a2^2*(4*a0*a2-a1^2)),-(a1*b2-a2*b1)/a2^2, + -sqrt(a1^2-4*a0*a2)*(a2*x+b2)/a2^2) + *%e^((sqrt(a1^2-4*a0*a2)*(a2*x+b2)-a1*a2*x)/(2*a2^2))]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 146 */ assume(x>0); @@ -273,27 +361,45 @@ /* Kamke 2. 149 */ (pn_(149),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x+b)*y,y,x)); -false; +[y = %k2*bessel_y(sqrt(1-4*b),2*sqrt(a)*sqrt(x))*sqrt(x) + +%k1*bessel_j(sqrt(1-4*b),2*sqrt(a)*sqrt(x))*sqrt(x)]; +method; +odelin; /* Kamke 2. 150 */ (pn_(150),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(x^2-2)*y,y,x)); -false; +[y = bessel_y(-3/2,-x)*%k2*sqrt(-x)+bessel_j(-3/2,-x)*%k1*sqrt(-x)]; +method; +odelin; /* Kamke 2. 151 */ (pn_(151),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-(a*x^2+2)*y,y,x)); -false; +[y = bessel_y(-3/2,-sqrt(-a)*x)*%k2*sqrt(-x) + +bessel_j(-3/2,-sqrt(-a)*x)*%k1*sqrt(-x)]; +method; +odelin; /* Kamke 2. 152 */ (pn_(152),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a^2*x^2-6)*y,y,x)); -false; +[y = bessel_y(5/2,-a*x)*%k2*sqrt(-x)+bessel_j(5/2,-a*x)*%k1*sqrt(-x)]; +method; +odelin; /* Kamke 2. 153 */ (pn_(153),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x^2-(v-1)*v)*y,y,x)); -false; +[y = %k2*bessel_y(-(2*v-1)/2,-sqrt(a)*x)*sqrt(-x) + +%k1*bessel_j(-(2*v-1)/2,-sqrt(a)*x)*sqrt(-x)]; +method; +odelin; /* Kamke 2. 154 */ (pn_(154),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x^2+b*x+c)*y,y,x)); -false; +[y = %k2*kummer_u((a*sqrt(1-4*c)-sqrt(-a)*b+a)/(2*a),sqrt(1-4*c)+1, + -2*sqrt(-a)*x)*(-x)^((sqrt(1-4*c)+1)/2)*%e^(sqrt(-a)*x) + +%k1*kummer_m((a*sqrt(1-4*c)-sqrt(-a)*b+a)/(2*a),sqrt(1-4*c)+1, + -2*sqrt(-a)*x)*(-x)^((sqrt(1-4*c)+1)/2)*%e^(sqrt(-a)*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 155 */ (pn_(155),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x^k-(b-1)*b)*y,y,x)); @@ -309,7 +415,9 @@ /* Kamke 2. 158 */ (pn_(158),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*'diff(y,x,1)+(-b^2*x^2-a*b)*y,y,x)); -false; +[y = %k2*%e^(b*x)*'integrate(%e^(a/x-2*b*x),x)+%k1*%e^(b*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 159 */ (pn_(159),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*'diff(y,x,1)-y-a*x^2,y,x)); @@ -333,7 +441,9 @@ /* Kamke 2. 161 */ (pn_(161),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*'diff(y,x,1)+(-x-a)*y,y,x)); -false; +[y=%k2*bessel_y(2*sqrt(a),2*sqrt(-x))+%k1*bessel_j(2*sqrt(a),2*sqrt(-x))]; +method; +odelin; /* Kamke 2. 162 */ declare(v,integer); @@ -357,7 +467,9 @@ /* Kamke 2. 164 */ (pn_(164),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*'diff(y,x,1)+(l*x^2-v^2)*y,y,x)); -false; +[y = %k2*bessel_y(v,-sqrt(l)*x)+%k1*bessel_j(v,-sqrt(l)*x)]; +method; +odelin; /* Kamke 2. 165 */ (pn_(165),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(x+a)*'diff(y,x,1)-y,y,x)); @@ -389,23 +501,41 @@ /* Kamke 2. 169 */ (pn_(169),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x*'diff(y,x,1)+(a*x-b^2)*y,y,x)); -false; +[y = %k2*bessel_y(sqrt(4*b^2+1),2*sqrt(a)*sqrt(x))/sqrt(x) + +%k1*bessel_j(sqrt(4*b^2+1),2*sqrt(a)*sqrt(x))/sqrt(x)]; +method; +odelin; /* Kamke 2. 170 */ (pn_(170),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x*'diff(y,x,1)+(a*x^2+b)*y,y,x)); -false; +[y = %k2*bessel_y(sqrt(1-4*b)/2,-sqrt(a)*x)*sqrt(-x)/x + +%k1*bessel_j(sqrt(1-4*b)/2,-sqrt(a)*x)*sqrt(-x)/x]; +method; +odelin; /* Kamke 2. 171 */ (pn_(171),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x*'diff(y,x,1)+(l*x^2+a*x-n*(n+1))*y,y,x)); -false; +[y = %k2*kummer_u(-(2*l*n+a*sqrt(-l))/(2*l),-2*n,-2*sqrt(-l)*x) + *%e^(sqrt(-l)*x)/((-x)^n*x) + +%k1*kummer_m(-(2*l*n+a*sqrt(-l))/(2*l),-2*n,-2*sqrt(-l)*x) + *%e^(sqrt(-l)*x)/((-x)^n*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; + /* Kamke 2. 172 */ (pn_(172),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*(x-1)*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*bessel_y(sqrt(1-4*a)/2,-%i/x)*%e^-(1/x)/sqrt(-x) + +%k1*bessel_j(sqrt(1-4*a)/2,-%i/x)*%e^-(1/x)/sqrt(-x)]; +method; +odelin; /* Kamke 2. 173 */ (pn_(173),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*(x+a)*'diff(y,x,1)-(b-1)*b*y,y,x)); -false; + [y = %k2*bessel_y(-(2*b-1)/2,-%i*a/x)*%e^(a/x)/sqrt(-x) + +%k1*bessel_j(-(2*b-1)/2,-%i*a/x)*%e^(a/x)/sqrt(-x)]; +method; +odelin; /* Kamke 2. 174 */ (pn_(174),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*'diff(y,x,1)+2*y-x^5*log(x),y,x)); @@ -425,7 +555,9 @@ /* Kamke 2. 176 */ (pn_(176),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*'diff(y,x,1)+(x^2+2)*y,y,x)); -false; +[y = %k2*x*%e^(%i*x)+%k1*x*%e^-(%i*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 177 */ (pn_(177),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(x^2+2)*'diff(y,x,1)-2*x*'diff(y,x,1)-x^2/cos(x),y,x)); @@ -441,7 +573,9 @@ /* Kamke 2. 179 */ (pn_(179),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*'diff(y,x,1)+(a^2*x^2+2)*y,y,x)); -false; +[y = %k2*x*%e^(%i*a*x)+%k1*x*%e^-(%i*a*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 180 */ (pn_(180),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+3*x*'diff(y,x,1)+(x^2-v^2+1)*y-f(x),y,x)); @@ -481,7 +615,10 @@ /* Kamke 2. 185 */ (pn_(185),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+5*x*'diff(y,x,1)+(4-2*x^3)*y,y,x)); -false; +[y = bessel_y(0,-2*sqrt(2)*sqrt(-x)*x/3)*%k2/x^2 + +bessel_j(0,-2*sqrt(2)*sqrt(-x)*x/3)*%k1/x^2]; +method; +odelin; /* Kamke 2. 186 */ (pn_(186),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-5*x*'diff(y,x,1)+8*y-x^3*sin(x),y,x)); @@ -505,7 +642,12 @@ /* Kamke 2. 188 */ (pn_(188),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*kummer_u(-(sqrt(-4*c+a^2-2*a+1)+a-3)/2,1-sqrt(-4*c+a^2-2*a+1), + -b/x)*x^((sqrt(-4*c+a^2-2*a+1)-a+1)/2)*%e^(b/x) + +%k1*kummer_m(-(sqrt(-4*c+a^2-2*a+1)+a-3)/2,1-sqrt(-4*c+a^2-2*a+1), + -b/x)*x^((sqrt(-4*c+a^2-2*a+1)-a+1)/2)*%e^(b/x)]; +method; +odelin; /* Kamke 2. 189 */ (pn_(189),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*x*'diff(y,x,1)+(b*x^m+c)*y,y,x)); @@ -513,45 +655,75 @@ /* Kamke 2. 190 */ (pn_(190),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x^2*'diff(y,x,1)+(a*x+b)*y,y,x)); -false; +[y = %k2*kummer_u((sqrt(1-4*b)+2*a+1)/2,sqrt(1-4*b)+1,-x) + *(-x)^((sqrt(1-4*b)+1)/2) + +%k1*kummer_m((sqrt(1-4*b)+2*a+1)/2,sqrt(1-4*b)+1,-x) + *(-x)^((sqrt(1-4*b)+1)/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 191 */ (pn_(191),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x^2*'diff(y,x,1)-2*y,y,x)); -false; +[y = bessel_y(-3/2,-%i*x/2)*%k2*sqrt(-x)*%e^-(x/2) + +bessel_j(-3/2,-%i*x/2)*%k1*sqrt(-x)*%e^-(x/2)]; +method; +odelin; /* Kamke 2. 192 */ (pn_(192),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(x^2-1)*'diff(y,x,1)-y,y,x)); -false; + [y = %k2*%e^-x*'integrate(%e^(x-1/x),x)+%k1*%e^-x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 193 */ (pn_(193),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x+1)*'diff(y,x,1)+(x-9)*y,y,x)); -false; +[y = kummer_u(-2,-5,-x)*%k2/x^3+kummer_m(-2,-5,-x)*%k1/x^3]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 194 */ (pn_(194),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x+1)*'diff(y,x,1)+(3*x-1)*y,y,x)); -false; +[y = kummer_u(4,3,-x)*%k2*x+kummer_m(4,3,-x)*%k1*x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 195 */ (pn_(195),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x+3)*'diff(y,x,1)-y,y,x)); -false; +[y = kummer_u(sqrt(2)-1,2*sqrt(2)+1,-x)*%k2*x^(sqrt(2)-1) + +kummer_m(sqrt(2)-1,2*sqrt(2)+1,-x)*%k1*x^(sqrt(2)-1)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; + /* Kamke 2. 196 */ (pn_(196),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-(x-1)*x*'diff(y,x,1)+(x-1)*y,y,x)); -false; +[y = %k2*x*'integrate(%e^x/x^3,x)+%k1*x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 197 */ (pn_(197),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-(x^2-2*x)*'diff(y,x,1)+(-x-a)*y,y,x)); -false; +[y = %k2*bessel_y(sqrt(4*a+1)/2,-%i*x/2)*sqrt(-x)*%e^(x/2)/x + +%k1*bessel_j(sqrt(4*a+1)/2,-%i*x/2)*sqrt(-x)*%e^(x/2)/x]; +method; +odelin; /* Kamke 2. 198 */ (pn_(198),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-(x^2-2*x)*'diff(y,x,1)+(-3*x-2)*y,y,x)); -false; +[y = %k2*x*%e^x*'integrate(%e^-x/x^4,x)+%k1*x*%e^x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 199 */ (pn_(199),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-x*(x+4)*'diff(y,x,1)+4*y,y,x)); -false; +[y = %k2*x^4*%e^x*'integrate(%e^-x/x^4,x)+%k1*x^4*%e^x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 200 */ (pn_(200),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x^2*'diff(y,x,1)-(v-1)*v*y,y,x)); -false; +[y = %k2*bessel_y(-(2*v-1)/2,-%i*x)*sqrt(-x)*%e^-x + +%k1*bessel_j(-(2*v-1)/2,-%i*x)*sqrt(-x)*%e^-x]; +method; +odelin; Index: rtestode_kamke_2_3.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_2_3.mac,v retrieving revision 1.3 retrieving revision 1.4 diff -u -d -r1.3 -r1.4 --- rtestode_kamke_2_3.mac 2 Sep 2006 12:22:36 -0000 1.3 +++ rtestode_kamke_2_3.mac 31 Jan 2007 12:44:12 -0000 1.4 @@ -11,19 +11,29 @@ /* Kamke 2. 201 */ (pn_(201),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(2*x+1)*'diff(y,x,1)-4*y,y,x)); -false; +[y = kummer_u(-2,-3,-2*x)*%k2/x^2+kummer_m(-2,-3,-2*x)*%k1/x^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 202 */ (pn_(202),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*(x+1)*'diff(y,x,1)+2*(x+1)*y,y,x)); -false; +[y = %k2*x*%e^(2*x)+%k1*x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 203 */ (pn_(203),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*x^2*'diff(y,x,1)-2*y,y,x)); -false; +[y = bessel_y(-3/2,-%i*a*x/2)*%k2*sqrt(-x)*%e^-(a*x/2) + +bessel_j(-3/2,-%i*a*x/2)*%k1*sqrt(-x)*%e^-(a*x/2)]; +method; +odelin; /* Kamke 2. 204 */ (pn_(204),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(2*b+a)*x^2*'diff(y,x,1)+(b*(b+a)*x^2-2)*y,y,x)); -false; +[y = bessel_y(-3/2,-%i*a*x/2)*%k2*sqrt(-x)*%e^-(b*x+a*x/2) + +bessel_j(-3/2,-%i*a*x/2)*%k1*sqrt(-x)*%e^-(b*x+a*x/2)]; +method; +odelin; /* Kamke 2. 205 */ (pn_(205),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*x^2*'diff(y,x,1)+f(x)*y,y,x)); @@ -31,27 +41,57 @@ /* Kamke 2. 206 */ (pn_(206),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(2*a*x+b)*'diff(y,x,1)+(c*x^2+a*b*x+d)*y,y,x)); -false; +[y=%k2*bessel_y(sqrt(-4*d+b^2-2*b+1)/2,-sqrt(c-a^2)*x)*%e^-(a*x)/x^(b/2-1/2) + +%k1*bessel_j(sqrt(-4*d+b^2-2*b+1)/2,-sqrt(c-a^2)*x)*%e^-(a*x)/x^(b/2-1/2)]; +method; +odelin; /* Kamke 2. 207 */ (pn_(207),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(a*x+b)*'diff(y,x,1)+(a1*x^2+b1*x+c_1)*y,y,x)); -false; +[y = %k2*kummer_u(((4*a1-a^2)*sqrt(-4*c_1+b^2-2*b+1) + +sqrt(a^2-4*a1)*(a*b-2*b1)+4*a1-a^2) + /(2*(4*a1-a^2)),sqrt(-4*c_1+b^2-2*b+1)+1, + -sqrt(a^2-4*a1)*x)*x^((sqrt(-4*c_1+b^2-2*b+1)-b+1)/2) + *%e^((sqrt(a^2-4*a1)-a)*x/2) + +%k1*kummer_m(((4*a1-a^2)*sqrt(-4*c_1+b^2-2*b+1) + +sqrt(a^2-4*a1)*(a*b-2*b1)+4*a1-a^2) + /(2*(4*a1-a^2)),sqrt(-4*c_1+b^2-2*b+1)+1, + -sqrt(a^2-4*a1)*x)*x^((sqrt(-4*c_1+b^2-2*b+1)-b+1)/2) + *%e^((sqrt(a^2-4*a1)-a)*x/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 208 */ (pn_(208),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x^3*'diff(y,x,1)+(x^2-2)*y,y,x)); -false; +[y = %k2*%e^-(x^2/2)*(sqrt(2)*sqrt(%pi)*%e^(x^2/2)*erf(x/sqrt(2))-2*x)/x + +%k1/x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 209 */ (pn_(209),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x^2+2)*'diff(y,x,1)+(x^2-2)*y,y,x)); -false; +[y = %k2*%e^-(x^2/2) + *(sqrt(2)*sqrt(%pi)*%i*erf(%i*x/sqrt(2))+2*x*%e^(x^2/2))/x^2 + +%k1*%e^-(x^2/2)/x^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 210 */ (pn_(210),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*(x^2-a)*'diff(y,x,1)+(2*n*x^2+a*((-1)^n-1))*y,y,x)); -false; +[y = %k2*kummer_u((sqrt(-4*a*(-1)^n+4*a^2+1)-2*n-2*a+1)/4, + (sqrt(-4*a*(-1)^n+4*a^2+1)+2)/2,x^2) + *x^((sqrt(-4*a*(-1)^n+4*a^2+1)-2*a+1)/2) + +%k1*kummer_m((sqrt(-4*a*(-1)^n+4*a^2+1)-2*n-2*a+1)/4, + (sqrt(-4*a*(-1)^n+4*a^2+1)+2)/2,x^2) + *x^((sqrt(-4*a*(-1)^n+4*a^2+1)-2*a+1)/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 211 */ (pn_(211),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+4*x^3*'diff(y,x,1)+(4*x^4+2*x^2+1)*y,y,x)); -false; +[y = %k2*x^((sqrt(-3)+1)/2)*%e^-x^2+%k1*%e^-x^2/x^(sqrt(-3)/2-1/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 212 */ (pn_(212),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(a*x^2+b)*'diff(y,x,1)+f(x)*y,y,x)); @@ -59,11 +99,21 @@ /* Kamke 2. 213 */ (pn_(213),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x^3+1)*'diff(y,x,1)-y,y,x)); -false; + [y = kummer_u(2/3,1/3,x^3/3)*%k2*%e^-(x^3/3)/x + +kummer_m(2/3,1/3,x^3/3)*%k1*%e^-(x^3/3)/x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 214 */ (pn_(214),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(-x^4+(2*n+2*a+1)*x^2+a*(-1)^n-a^2)*y,y,x)); -false; +[y = %k2*kummer_u((sqrt(-4*a*(-1)^n+4*a^2+1)-2*n-2*a+1)/4, + (sqrt(-4*a*(-1)^n+4*a^2+1)+2)/2,x^2) + *x^((sqrt(-4*a*(-1)^n+4*a^2+1)+1)/2)*%e^-(x^2/2) + +%k1*kummer_m((sqrt(-4*a*(-1)^n+4*a^2+1)-2*n-2*a+1)/4, + (sqrt(-4*a*(-1)^n+4*a^2+1)+2)/2,x^2) + *x^((sqrt(-4*a*(-1)^n+4*a^2+1)+1)/2)*%e^-(x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 215 */ (pn_(215),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(a*x^n+b)*'diff(y,x,1)+(a1*x^(2*n)+b1*x^n+c_1)*y,y,x)); @@ -123,19 +173,34 @@ /* Kamke 2. 225 */ (pn_(225),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,2)-x*'diff(y,x,1)+y,y,x)); -false; +[y = gauss_b(2,2,5/2,-(%i*x-1)/2)*%k2*(-%i*x-1)^(1/4)*(1-%i*x)^(1/4) + *(x^2+1)^(5/4) + +gauss_a(2,2,5/2,-(%i*x-1)/2)*%k1*(-%i*x-1)^(1/4)*(1-%i*x)^(1/4) + *(x^2+1)^(5/4)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 226 */ (pn_(226),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,2)+2*x*'diff(y,x,1)-(v-1)*v*y,y,x)); -false; +[y = %k2*gauss_b(1-v,v,1,-(%i*x-1)/2)*sqrt(-%i*x-1)*sqrt(1-%i*x)/sqrt(x^2+1) + +%k1*gauss_a(1-v,v,1,-(%i*x-1)/2)*sqrt(-%i*x-1)*sqrt(1-%i*x)/sqrt(x^2+1)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 227 */ (pn_(227),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,2)-2*x*'diff(y,x,1)+2*y,y,x)); -false; +[y = %k2*(x^2+1)*%e^(2*%i*atan(x))+%k1*(x^2+1)*%e^-(2*%i*atan(x))]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 228 */ (pn_(228),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,2)+3*x*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*gauss_b(sqrt(1-a),-sqrt(1-a),1/2,-(%i*x-1)/2)*(-%i*x-1)^(1/4) + *(1-%i*x)^(1/4)/(x^2+1)^(3/4) + +%k1*gauss_a(sqrt(1-a),-sqrt(1-a),1/2,-(%i*x-1)/2)*(-%i*x-1)^(1/4) + *(1-%i*x)^(1/4)/(x^2+1)^(3/4)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 229 */ (pn_(229),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,2)+4*x*'diff(y,x,1)+2*y-2*cos(x)+2*x,y,x)); @@ -155,7 +220,9 @@ /* Kamke 2. 231 */ (pn_(231),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-v*(v+1)*y,y,x)); -false; +[y = %k2*gauss_b(-v-1,v,0,-(x-1)/2)+%k1*gauss_a(-v-1,v,0,-(x-1)/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 232 */ (pn_(232),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-n*(n+1)*y+Diff(LegendreP(n,x),x),y,x)); @@ -203,31 +270,55 @@ /* Kamke 2. 239 */ (pn_(239),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+2*x*'diff(y,x,1)-l*y,y,x)); -false; + [y = %k2*gauss_b((sqrt(4*l+1)+1)/2,-(sqrt(4*l+1)-1)/2,1,-(x-1)/2) + +%k1*gauss_a((sqrt(4*l+1)+1)/2,-(sqrt(4*l+1)-1)/2,1,-(x-1)/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 240 */ (pn_(240),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+2*x*'diff(y,x,1)-v*(v+1)*y,y,x)); -false; +[y = %k2*gauss_b(-v,v+1,1,-(x-1)/2)+%k1*gauss_a(-v,v+1,1,-(x-1)/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 241 */ (pn_(241),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-2*x*'diff(y,x,1)+(-v-2)*(v-1)*y,y,x)); -false; +[y = %k2*gauss_b(-v,v+1,3,-(x-1)/2)*(x-1)^2 + +%k1*gauss_a(-v,v+1,3,-(x-1)/2)*(x-1)^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 242 */ (pn_(242),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-(3*x+1)*'diff(y,x,1)+(x-x^2)*y,y,x)); -false; + [y = %k2*%e^-x + *(x^3*'integrate((8*x-7)*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1),x) + +3*x^2*'integrate((8*x-7)*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1),x) + +3*x*'integrate((8*x-7)*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1),x) + +'integrate((8*x-7)*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1),x) + -2*x^2*%e^(2*x)+3*x*%e^(2*x))/(x+1) + +%k1*(x+1)^2*%e^-x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 243 */ (pn_(243),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+4*x*'diff(y,x,1)+(x^2+1)*y,y,x)); -false; +[y = %k2*%e^(%i*x)/((x-1)*(x+1))+%k1*%e^-(%i*x)/((x-1)*(x+1))]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 244 */ (pn_(244),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+2*(n+1)*x*'diff(y,x,1)+(-v-n-1)*(v-n)*y,y,x)); -false; + [y = %k2*gauss_b(-v,v+1,n+1,-(x-1)/2)/(x+1)^n + +%k1*gauss_a(-v,v+1,n+1,-(x-1)/2)/(x+1)^n]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 245 */ (pn_(245),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-2*(n-1)*x*'diff(y,x,1)+(-v+n-1)*(v+n)*y,y,x)); -false; +[y = %k2*gauss_b(-v,v+1,n+1,-(x-1)/2)*(x-1)^n + +%k1*gauss_a(-v,v+1,n+1,-(x-1)/2)*(x-1)^n]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 246 */ (pn_(246),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-2*(v-1)*x*'diff(y,x,1)-2*v*y,y,x)); @@ -239,27 +330,43 @@ /* Kamke 2. 247 */ (pn_(247),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+2*a*x*'diff(y,x,1)+(a-1)*a*y,y,x)); -false; + [y = %k2*(x+1)^(1-a)*'integrate((x+1)^a/((x-1)^a*(x^2+2*x+1)),x) + +%k1*(x+1)^(1-a)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 248 */ +/* Take a long time (pn_(248),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+a*x*'diff(y,x,1)+(b*x^2+c*x+d)*y,y,x)); false; +*/ /* Kamke 2. 249 */ +/* Take a long time (pn_(249),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+c*y,y,x)); false; +*/ /* Kamke 2. 250 */ (pn_(250),ans:contrib_ode(eqn:(x^2-a^2)*'diff(y,x,2)+8*x*'diff(y,x,1)+12*y,y,x)); -false; +[y = %k2*(3*x^2+a^2)/((x-a)^3*(x+a)^3)+%k1/(x+a)^3]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 251 */ (pn_(251),ans:contrib_ode(eqn:x*(x+1)*'diff(y,x,2)-(x-1)*'diff(y,x,1)+y,y,x)); -false; +[y = gauss_b(2,2,1,-x)*%k2*(x+1)^3+gauss_a(2,2,1,-x)*%k1*(x+1)^3]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 252 */ (pn_(252),ans:contrib_ode(eqn:x*(x+1)*'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*gauss_b((sqrt(-4*c+a^2-2*a+1)+a-1)/2, + -(sqrt(-4*c+a^2-2*a+1)-a+1)/2,b,-x) + +%k1*gauss_a((sqrt(-4*c+a^2-2*a+1)+a-1)/2, + -(sqrt(-4*c+a^2-2*a+1)-a+1)/2,b,-x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 253 */ (pn_(253),ans:contrib_ode(eqn:x*(x+1)*'diff(y,x,2)+(3*x+2)*'diff(y,x,1)+y,y,x)); @@ -271,7 +378,16 @@ /* Kamke 2. 254 */ (pn_(254),ans:contrib_ode(eqn:(x^2+x-2)*'diff(y,x,2)+(x^2-x)*'diff(y,x,1)+(-6*x^2-7*x)*y,y,x)); -false; + [y = %k2*%e^-(3*x) + *(2*x^2*%e^(5*x) + *'integrate((7*x-12)*%e^-(5*x)/(x^3-3*x^2+3*x-1),x) + -4*x*%e^(5*x) + *'integrate((7*x-12)*%e^-(5*x)/(x^3-3*x^2+3*x-1),x) + +2*%e^(5*x)*'integrate((7*x-12)*%e^-(5*x)/(x^3-3*x^2+3*x-1),x) + -x^2-4*x)/(x-1) + +%k1*(x-1)*%e^(2*x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 255 */ (pn_(255),ans:contrib_ode(eqn:(x-1)*x*'diff(y,x,2)+a*'diff(y,x,1)-2*y,y,x)); @@ -283,7 +399,9 @@ /* Kamke 2. 256 */ (pn_(256),ans:contrib_ode(eqn:(x-1)*x*'diff(y,x,2)+(2*x-1)*'diff(y,x,1)-v*(v+1)*y,y,x)); -false; +[y = %k2*gauss_b(-v,v+1,1,x)+%k1*gauss_a(-v,v+1,1,x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 257 */ (pn_(257),ans:contrib_ode(eqn:(x-1)*x*'diff(y,x,2)+((a+1)*x+b)*'diff(y,x,1),y,x)); @@ -295,11 +413,19 @@ /* Kamke 2. 258 */ (pn_(258),ans:contrib_ode(eqn:(x-1)*x*'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*gauss_b((sqrt(-4*c+a^2-2*a+1)+a-1)/2, + -(sqrt(-4*c+a^2-2*a+1)-a+1)/2,-b,x) + +%k1*gauss_a((sqrt(-4*c+a^2-2*a+1)+a-1)/2, + -(sqrt(-4*c+a^2-2*a+1)-a+1)/2,-b,x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 259 */ (pn_(259),ans:contrib_ode(eqn:(x-1)*x*'diff(y,x,2)+((a+1)*x+b)*'diff(y,x,1)-l*y,y,x)); -false; +[y = %k2*gauss_b((sqrt(4*l+a^2)+a)/2,-(sqrt(4*l+a^2)-a)/2,-b,x) + +%k1*gauss_a((sqrt(4*l+a^2)+a)/2,-(sqrt(4*l+a^2)-a)/2,-b,x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 260 */ (pn_(260),ans:contrib_ode(eqn:(x-1)*x*'diff(y,x,2)+((b1+a1+1)*x-d1)*'diff(y,x,1)+a1*b1*d1,y,x)); @@ -315,7 +441,10 @@ /* Kamke 2. 262 */ (pn_(262),ans:contrib_ode(eqn:(x+1)^2*'diff(y,x,2)+(x^2+x-1)*'diff(y,x,1)+(-x-2)*y,y,x)); -false; +[y = %k2*%e^-x*'integrate((x+1)*%e^(x^2/(x+1)+x/(x+1)-1/(x+1)),x) + +%k1*%e^-x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 263 CHECKME - seems OK */ (pn_(263),ans:contrib_ode(eqn:x*(x+3)*'diff(y,x,2)+(3*x-1)*'diff(y,x,1)+y+(-20*x-30)*(x^2+3*x)^(7/3),y,x)); @@ -327,7 +456,9 @@ /* Kamke 2. 264 */ (pn_(264),ans:contrib_ode(eqn:(x^2+3*x+4)*'diff(y,x,2)+(x^2+x+1)*'diff(y,x,1)+(-2*x-3)*y,y,x)); -false; +[y = %k2*%e^-x+%k1*(x^2+x+3)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 265 */ (pn_(265),ans:contrib_ode(eqn:(x-2)*(x-1)*'diff(y,x,2)-(2*x-3)*'diff(y,x,1)+y,y,x)); @@ -355,11 +486,30 @@ /* Kamke 2. 269 */ (pn_(269),ans:contrib_ode(eqn:2*(x-1)*x*'diff(y,x,2)+((2*v+5)*x-2*v-3)*'diff(y,x,1)+(v+1)*y,y,x)); -false; +[y = %k2*gauss_b(-v,1/2,-(2*v-1)/2,x)/x^(v+1/2) + +%k1*gauss_a(-v,1/2,-(2*v-1)/2,x)/x^(v+1/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 270 */ (pn_(270),ans:contrib_ode(eqn:(2*x^2+6*x+4)*'diff(y,x,2)+(10*x^2+21*x+8)*'diff(y,x,1)+(12*x^2+17*x+8)*y,y,x)); -false; +[y = %k2*%e^-(3*x) + *(x^5*sqrt(x+1)*'integrate((43*x+36)*%e^x + /(sqrt(x+1)*(x^6+12*x^5+60*x^4+160*x^3+240*x^2+192*x+64)),x) + +10*x^4*sqrt(x+1)*'integrate((43*x+36)*%e^x + /(sqrt(x+1)*(x^6+12*x^5+60*x^4+160*x^3+240*x^2+192*x+64)),x) + +40*x^3*sqrt(x+1)*'integrate((43*x+36)*%e^x + /(sqrt(x+1)*(x^6+12*x^5+60*x^4+160*x^3+240*x^2+192*x+64)),x) + +80*x^2*sqrt(x+1)*'integrate((43*x+36)*%e^x + /(sqrt(x+1)*(x^6+12*x^5+60*x^4+160*x^3+240*x^2+192*x+64)),x) + +80*x*sqrt(x+1)*'integrate((43*x+36)*%e^x + /(sqrt(x+1)*(x^6+12*x^5+60*x^4+160*x^3+240*x^2+192*x+64)),x) + +32*sqrt(x+1)*'integrate((43*x+36)*%e^x + /(sqrt(x+1)*(x^6+12*x^5+60*x^4+160*x^3+240*x^2+192*x+64)),x) + +4*x^2*%e^x+22*x*%e^x+18*%e^x)/(sqrt(x+1)*(x+2)) + +%k1*(x+2)^4*%e^-(3*x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 271 */ (pn_(271),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+y,y,x)); @@ -371,15 +521,22 @@ /* Kamke 2. 272 */ (pn_(272),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+(4*a^2*x^2+1)*y,y,x)); -false; +[y = bessel_y(0,-a*x)*%k2*sqrt(-x)+bessel_j(0,-a*x)*%k1*sqrt(-x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 273 */ (pn_(273),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)-(x^2-4*k*x+4*m^2-1)*y,y,x)); -false; +[y = %k2*kummer_u(-(2*m-2*k-1)/2,1-2*m,-x)*%e^(x/2)/(-x)^(m-1/2) + +%k1*kummer_m(-(2*m-2*k-1)/2,1-2*m,-x)*%e^(x/2)/(-x)^(m-1/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 274 */ (pn_(274),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+4*x*'diff(y,x,1)+(x-v^2)*y,y,x)); -false; +[y = %k2*bessel_y(v,sqrt(x))+%k1*bessel_j(v,sqrt(x))]; +method; +odelin; /* Kamke 2. 275 */ (pn_(275),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+4*x*'diff(y,x,1)+(-x^2+2*(-m+2*l+1)*x-m^2+1)*y,y,x)); @@ -391,7 +548,9 @@ /* Kamke 2. 277 */ (pn_(277),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+4*x*'diff(y,x,1)+(-a*x^2-1)*y,y,x)); -false; +[y = %k2*%e^(sqrt(a)*x/2)/sqrt(x)+%k1*%e^-(sqrt(a)*x/2)/sqrt(x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 278 */ (pn_(278),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+4*x*'diff(y,x,1)+f(x)*y,y,x)); @@ -407,15 +566,21 @@ /* Kamke 2. 280 */ (pn_(280),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+8*x*'diff(y,x,1)+(-4*x^2-12*x-3)*y,y,x)); -false; +[y = %k2*sqrt(x)*%e^x*'integrate(%e^-(2*x)/x^3,x)+%k1*sqrt(x)*%e^x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 281 */ (pn_(281),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)-4*x*(2*x-1)*'diff(y,x,1)+(4*x^2-4*x-1)*y,y,x)); -false; +[y = %k2*sqrt(x)*%e^x+%k1*%e^x/sqrt(x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 282 */ (pn_(282),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+4*x^3*'diff(y,x,1)+(x^2-4)*(x^2+6)*y,y,x)); -false; +[y = %k2*x^3*%e^-(x^2/4)+%k1*%e^-(x^2/4)/x^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 283 */ (pn_(283),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+4*x^2*log(x)*'diff(y,x,1)+(x^2*log(x)^2+2*x-8)*y-4*x^2*%e^(x/2)/sqrt(x^x),y,x)); @@ -431,7 +596,10 @@ /* Kamke 2. 285 */ (pn_(285),ans:contrib_ode(eqn:x*(4*x-1)*'diff(y,x,2)+((4*a+2)*x-a)*'diff(y,x,1)+(a-1)*a*y,y,x)); -false; +[y = %k2*gauss_b(a/2,(a+1)/2,a,4*x)*sqrt(4*x-1) + +%k1*gauss_a(a/2,(a+1)/2,a,4*x)*sqrt(4*x-1)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 286 */ (pn_(286),ans:contrib_ode(eqn:(3*x-1)^2*'diff(y,x,2)+3*(3*x-1)*'diff(y,x,1)-9*y-log(3*x-1)^2,y,x)); @@ -443,15 +611,23 @@ /* Kamke 2. 287 */ (pn_(287),ans:contrib_ode(eqn:9*(x-1)*x*'diff(y,x,2)+3*(2*x-1)*'diff(y,x,1)-20*y,y,x)); -false; +[y = gauss_b(-1,2,5/3,x)*%k2*x^(2/3)+gauss_a(-1,2,5/3,x)*%k1*x^(2/3)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 288 */ (pn_(288),ans:contrib_ode(eqn:16*x^2*'diff(y,x,2)+(4*x+3)*y,y,x)); -false; +[y = bessel_y(-1/2,sqrt(x))*%k2*sqrt(x) + +bessel_j(-1/2,sqrt(x))*%k1*sqrt(x)]; +method; +odelin; /* Kamke 2. 289 */ (pn_(289),ans:contrib_ode(eqn:16*x^2*'diff(y,x,2)+32*x*'diff(y,x,1)+(-4*x-5)*y,y,x)); -false; +[y = bessel_y(-3/2,sqrt(-x))*%k2*sqrt(-x)/x + +bessel_j(-3/2,sqrt(-x))*%k1*sqrt(-x)/x]; +method; +odelin /* Kamke 2. 290 */ assume(x>0); @@ -475,11 +651,17 @@ /* Kamke 2. 293 */ (pn_(293),ans:contrib_ode(eqn:144*(x-1)*x*'diff(y,x,2)+(120*x-48)*'diff(y,x,1)+y,y,x)); -false; +[y = gauss_b(7/12,7/12,5/3,x)*%k2*x^(2/3) + +gauss_a(7/12,7/12,5/3,x)*%k1*x^(2/3)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 294 */ (pn_(294),ans:contrib_ode(eqn:144*(x-1)*x*'diff(y,x,2)+(168*x-96)*'diff(y,x,1)+y,y,x)); -false; +[y = gauss_b(5/12,5/12,4/3,x)*%k2*x^(1/3) + +gauss_a(5/12,5/12,4/3,x)*%k1*x^(1/3)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 295 */ (pn_(295),ans:contrib_ode(eqn:a*x^2*'diff(y,x,2)+b*x*'diff(y,x,1)+(c*x^2+d*x+f)*y,y,x)); @@ -487,7 +669,24 @@ /* Kamke 2. 296 */ (pn_(296),ans:contrib_ode(eqn:a2*x^2*'diff(y,x,2)+(a1*x^2+b1*x)*'diff(y,x,1)+(a0*x^2+b0*x+c_0)*y,y,x)); -false; +[y = %k2*kummer_u(((4*a0*a2-a1^2)*sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2) + +sqrt(a1^2-4*a0*a2)*(a1*b1-2*a2*b0)+4*a0*a2^2 + -a1^2*a2) + /(2*a2*(4*a0*a2-a1^2)), + (sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2)+a2)/a2, + -sqrt(a1^2-4*a0*a2)*x/a2) + *x^((sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2)-b1+a2)/(2*a2)) + *%e^((sqrt(a1^2-4*a0*a2)-a1)*x/(2*a2)) + +%k1*kummer_m(((4*a0*a2-a1^2)*sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2) + +sqrt(a1^2-4*a0*a2)*(a1*b1-2*a2*b0)+4*a0*a2^2 + -a1^2*a2) + /(2*a2*(4*a0*a2-a1^2)), + (sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2)+a2)/a2, + -sqrt(a1^2-4*a0*a2)*x/a2) + *x^((sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2)-b1+a2)/(2*a2)) + *%e^((sqrt(a1^2-4*a0*a2)-a1)*x/(2*a2))]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 297 */ assume(a>0); @@ -503,7 +702,18 @@ /* Kamke 2. 298 */ (pn_(298),ans:contrib_ode(eqn:(a*x^2+1)*'diff(y,x,2)+b*x*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*gauss_b((sqrt(-4*a*c+b^2-2*a*b+a^2)+a)/(2*a), + -(sqrt(-4*a*c+b^2-2*a*b+a^2)-a)/(2*a),b/(2*a), + -(sqrt(-a)*x-1)/2)*(1-sqrt(-a)*x)^(b/(4*a)) + *(sqrt(-a)*x+1) + /((-sqrt(-a)*x-1)^(b/(4*a))*(a*x^2+1)^(b/(4*a))) + +%k1*gauss_a((sqrt(-4*a*c+b^2-2*a*b+a^2)+a)/(2*a), + -(sqrt(-4*a*c+b^2-2*a*b+a^2)-a)/(2*a),b/(2*a), + -(sqrt(-a)*x-1)/2)*(1-sqrt(-a)*x)^(b/(4*a)) + *(sqrt(-a)*x+1) + /((-sqrt(-a)*x-1)^(b/(4*a))*(a*x^2+1)^(b/(4*a)))]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 299 */ (pn_(299),ans:contrib_ode(eqn:(a^2*x^2-1)*'diff(y,x,2)+2*a^2*x*'diff(y,x,1),y,x)); @@ -515,5 +725,7 @@ /* Kamke 2. 300 */ (pn_(300),ans:contrib_ode(eqn:(a^2*x^2-1)*'diff(y,x,2)+2*a^2*x*'diff(y,x,1)-2*a^2*y,y,x)); -false; +[y = gauss_b(-1,2,1,-(a*x-1)/2)*%k2+gauss_a(-1,2,1,-(a*x-1)/2)*%k1]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; ```

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 [Maxima-commits] CVS: maxima/share/contrib/diffequations/tests rtestode_kamke_2_1.mac, 1.2, 1.3 rtestode_kamke_2_2.mac, 1.4, 1.5 rtestode_kamke_2_3.mac, 1.3, 1.4 From: David Billinghurst - 2007-01-31 12:44:22 ```Update of /cvsroot/maxima/maxima/share/contrib/diffequations/tests In directory sc8-pr-cvs7.sourceforge.net:/tmp/cvs-serv18653 Modified Files: rtestode_kamke_2_1.mac rtestode_kamke_2_2.mac rtestode_kamke_2_3.mac Log Message: odelin solves many 2nd order linear ODEs. Update testsuite. Index: rtestode_kamke_2_1.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_2_1.mac,v retrieving revision 1.2 retrieving revision 1.3 diff -u -d -r1.2 -r1.3 --- rtestode_kamke_2_1.mac 2 Sep 2006 12:22:36 -0000 1.2 +++ rtestode_kamke_2_1.mac 31 Jan 2007 12:44:11 -0000 1.3 @@ -93,19 +93,29 @@ /* Kamke 2. 10 */ (pn_(10),ans:contrib_ode(eqn:'diff(y,x,2)+(a*x+b)*y,y,x)); -false; +[y = bessel_y(1/3,2*(a*x+b)^(3/2)/(3*a))*%k2*sqrt(a*x+b) + +bessel_j(1/3,2*(a*x+b)^(3/2)/(3*a))*%k1*sqrt(a*x+b)]; +method; +odelin; /* Kamke 2. 11 */ (pn_(11),ans:contrib_ode(eqn:'diff(y,x,2)-(x^2+1)*y,y,x)); -false; +[y = %k2*%e^(x^2/2)*erf(x)+%k1*%e^(x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 12 */ (pn_(12),ans:contrib_ode(eqn:'diff(y,x,2)-(x^2+a)*y,y,x)); -false; +[y = %k2*kummer_u(-(a-1)/4,1/2,-x^2)*%e^(x^2/2) + +%k1*kummer_m(-(a-1)/4,1/2,-x^2)*%e^(x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 13 */ (pn_(13),ans:contrib_ode(eqn:'diff(y,x,2)-(a^2*x^2+a)*y,y,x)); -false; +[y = %k2*%e^(a*x^2/2)*erf(sqrt(a)*x)+%k1*%e^(a*x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 14 */ (pn_(14),ans:contrib_ode(eqn:'diff(y,x,2)-c*x^a*y,y,x)); @@ -225,7 +235,10 @@ /* Kamke 2. 37 */ (pn_(37),ans:contrib_ode(eqn:'diff(y,x,2)+a*'diff(y,x,1)+(-b^2*x^2-c)*y,y,x)); -false; +[y = %k2*kummer_u(-(4*c-4*b+a^2)/(16*b),1/2,-b*x^2)*%e^(x*(b*x-a)/2) + +%k1*kummer_m(-(4*c-4*b+a^2)/(16*b),1/2,-b*x^2)*%e^(x*(b*x-a)/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 38 */ (pn_(38),ans:contrib_ode(eqn:'diff(y,x,2)+2*a*'diff(y,x,1)+f(x)*y,y,x)); @@ -241,39 +254,61 @@ /* Kamke 2. 40 */ (pn_(40),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)-y,y,x)); -false; +[y = kummer_u(0,3/2,-x^2/2)*%k2*x+kummer_m(0,3/2,-x^2/2)*%k1*x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 41 */ (pn_(41),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)+(n+1)*y,y,x)); -false; +[y = %k2*kummer_u((n+1)/2,1/2,-x^2/2)+%k1*kummer_m((n+1)/2,1/2,-x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 42 */ (pn_(42),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)-n*y,y,x)); -false; +[y = %k2*kummer_u(-n/2,1/2,-x^2/2)+%k1*kummer_m(-n/2,1/2,-x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 43 */ (pn_(43),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)+2*y,y,x)); -false; +[y = kummer_u(2,3/2,-x^2/2)*%k2*x*%e^(x^2/2) + +kummer_m(2,3/2,-x^2/2)*%k1*x*%e^(x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 44 */ (pn_(44),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)-a*y,y,x)); -false; +[y = %k2*kummer_u(-(a-1)/2,1/2,-x^2/2)*%e^(x^2/2) + +%k1*kummer_m(-(a-1)/2,1/2,-x^2/2)*%e^(x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 45 */ (pn_(45),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)+(x-1)*y,y,x)); -false; +[y = %k2*%e^x+%k1*erf(%i*(x-2)/sqrt(2))*%e^(x-2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 46 */ (pn_(46),ans:contrib_ode(eqn:'diff(y,x,2)-2*x*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*kummer_u((a+2)/4,1/2,-x^2)*%e^x^2 + +%k1*kummer_m((a+2)/4,1/2,-x^2)*%e^x^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 47 */ (pn_(47),ans:contrib_ode(eqn:'diff(y,x,2)+4*x*'diff(y,x,1)+(4*x^2+2)*y,y,x)); -false; +[y = %k2*x*%e^-x^2+%k1*%e^-x^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 48 */ (pn_(48),ans:contrib_ode(eqn:'diff(y,x,2)-4*x*'diff(y,x,1)+(3*x^2+2*n-1)*y,y,x)); -false; +[y = %k2*kummer_u((n+1)/2,1/2,-x^2)*%e^(3*x^2/2) + +%k1*kummer_m((n+1)/2,1/2,-x^2)*%e^(3*x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 49 */ (pn_(49),ans:contrib_ode(eqn:'diff(y,x,2)-4*x*'diff(y,x,1)+(4*x^2-1)*y-%e^x,y,x)); @@ -281,7 +316,9 @@ /* Kamke 2. 50 */ (pn_(50),ans:contrib_ode(eqn:'diff(y,x,2)-4*x*'diff(y,x,1)+(4*x^2-2)*y,y,x)); -false; +[y = %k2*x*%e^x^2+%k1*%e^x^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 51 */ (pn_(51),ans:contrib_ode(eqn:'diff(y,x,2)-4*x*'diff(y,x,1)+(4*x^2-3)*y-%e^x^2,y,x)); @@ -289,39 +326,60 @@ /* Kamke 2. 52 */ (pn_(52),ans:contrib_ode(eqn:'diff(y,x,2)+a*x*'diff(y,x,1)+b*y,y,x)); -false; + [y = %k2*kummer_u(b/(2*a),1/2,-a*x^2/2) + +%k1*kummer_m(b/(2*a),1/2,-a*x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 53 */ (pn_(53),ans:contrib_ode(eqn:'diff(y,x,2)+2*a*x*'diff(y,x,1)+a^2*x^2*y,y,x)); -false; +[y = %k2*%e^-(a*x^2/2+sqrt(a)*x)+%k1*%e^(x*(2*sqrt(a)-a*x)/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 54 */ +/* CHECKME: Takes a long time (pn_(54),ans:contrib_ode(eqn:'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+(c*x+d)*y,y,x)); false; +*/ /* Kamke 2. 55 */ +/* CHECKME: Takes a long time (pn_(55),ans:contrib_ode(eqn:'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+(a1*x^2+b1*x+c_1)*y,y,x)); false; +*/ /* Kamke 2. 56 */ (pn_(56),ans:contrib_ode(eqn:'diff(y,x,2)-x^2*'diff(y,x,1)+x*y,y,x)); -false; +[y = kummer_u(1,2/3,-x^3/3)*%k2*%e^(x^3/3) + +kummer_m(1,2/3,-x^3/3)*%k1*%e^(x^3/3)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 57 */ (pn_(57),ans:contrib_ode(eqn:'diff(y,x,2)-x^2*'diff(y,x,1)-(x+1)^2*y,y,x)); -false; +[y = %k2*%e^(x*(x^2+3)/3)*'integrate(%e^(-x^3/3-2*x),x) + +%k1*%e^(x*(x^2+3)/3)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 58 */ (pn_(58),ans:contrib_ode(eqn:'diff(y,x,2)-x^2*(x+1)*'diff(y,x,1)+x*(x^4-2)*y,y,x)); -false; +[y = %k2*%e^(x^3/3)*'integrate(%e^(x^4/4-x^3/3),x)+%k1*%e^(x^3/3)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 59 */ (pn_(59),ans:contrib_ode(eqn:'diff(y,x,2)+x^4*'diff(y,x,1)-x^3*y,y,x)); -false; +[y = kummer_u(0,6/5,-x^5/5)*%k2*x+kummer_m(0,6/5,-x^5/5)*%k1*x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; -/* Kamke 2. 60 */ +/* Kamke 2. 60 */ +/* CHECKME: Takes a long time (pn_(60),ans:contrib_ode(eqn:'diff(y,x,2)+a*x^(q-1)*'diff(y,x,1)+b*x^(q-2)*y,y,x)); false; +*/ /* Kamke 2. 61 */ (pn_(61),ans:contrib_ode(eqn:'diff(y,x,2)+sqrt(x)*'diff(y,x,1)+(x/4+1/(4*sqrt(x))-9)*y-x*%e^-(x^(3/2)/3),y,x)); @@ -441,11 +499,16 @@ /* Kamke 2. 86 */ (pn_(86),ans:contrib_ode(eqn:4*'diff(y,x,2)+9*x*y,y,x)); -false; +[y = bessel_y(1/3,x^(3/2))*%k2*sqrt(x)+bessel_j(1/3,x^(3/2))*%k1*sqrt(x)]; +method; +odelin; /* Kamke 2. 87 */ (pn_(87),ans:contrib_ode(eqn:4*'diff(y,x,2)-(x^2+a)*y,y,x)); -false; +[y = %k2*kummer_u(-(a-2)/8,1/2,-x^2/2)*%e^(x^2/4) + +%k1*kummer_m(-(a-2)/8,1/2,-x^2/2)*%e^(x^2/4)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 88 */ (pn_(88),ans:contrib_ode(eqn:4*'diff(y,x,2)+4*tan(x)*'diff(y,x,1)+(-5*tan(x)^2-2)*y,y,x)); @@ -453,7 +516,12 @@ /* Kamke 2. 89 */ (pn_(89),ans:contrib_ode(eqn:a*'diff(y,x,2)-(x+c+a*b)*'diff(y,x,1)+(b*(x+c)+d)*y,y,x)); -false; +[y=%k2*kummer_u((d+1)/2,1/2,-(x^2+(2*c-2*a*b)*x+c^2-2*a*b*c+a^2*b^2)/(2*a)) + *%e^((2*x^2+4*c*x+c^2-2*a*b*c+a^2*b^2)/(4*a)) + +%k1*kummer_m((d+1)/2,1/2,-(x^2+(2*c-2*a*b)*x+c^2-2*a*b*c+a^2*b^2)/(2*a)) + *%e^((2*x^2+4*c*x+c^2-2*a*b*c+a^2*b^2)/(4*a))]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 90 */ (pn_(90),ans:contrib_ode(eqn:a^2*'diff(y,x,2)+a*(a^2-2*b*%e^-(a*x))*'diff(y,x,1)+b^2*%e^-(2*a*x)*y,y,x)); @@ -475,7 +543,10 @@ /* Kamke 2. 92 */ (pn_(92),ans:contrib_ode(eqn:x*'diff(y,x,2)+(x+a)*y,y,x)); -false; +[y = %k2*kummer_u(-%i*a/2,0,-2*%i*x)*%e^(%i*x) + +%k1*kummer_m(-%i*a/2,0,-2*%i*x)*%e^(%i*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 93 */ (pn_(93),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1),y,x)); @@ -487,19 +558,30 @@ /* Kamke 2. 94 */ (pn_(94),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1)+a*y,y,x)); -false; +[y = bessel_y(0,2*sqrt(a)*sqrt(x))*%k2 + +bessel_j(0,2*sqrt(a)*sqrt(x))*%k1]; +method; +odelin; /* Kamke 2. 95 */ (pn_(95),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1)+l*x*y,y,x)); -false; +[y = bessel_y(0,-sqrt(l)*x)*%k2+bessel_j(0,-sqrt(l)*x)*%k1]; +method; +odelin; /* Kamke 2. 96 */ (pn_(96),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1)+(x+a)*y,y,x)); -false; +[y = %k2*kummer_u(-(%i*a-1)/2,1,-2*%i*x)*%e^(%i*x) + +%k1*kummer_m(-(%i*a-1)/2,1,-2*%i*x)*%e^(%i*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 97 */ (pn_(97),ans:contrib_ode(eqn:x*'diff(y,x,2)-'diff(y,x,1)+a*y,y,x)); -false; +[y = bessel_y(-2,2*sqrt(a)*sqrt(x))*%k2*x + +bessel_j(-2,2*sqrt(a)*sqrt(x))*%k1*x]; +method; +odelin; /* Kamke 2. 98 */ assume(a>0); Index: rtestode_kamke_2_2.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_2_2.mac,v retrieving revision 1.4 retrieving revision 1.5 diff -u -d -r1.4 -r1.5 --- rtestode_kamke_2_2.mac 4 Sep 2006 12:27:16 -0000 1.4 +++ rtestode_kamke_2_2.mac 31 Jan 2007 12:44:11 -0000 1.5 @@ -11,23 +11,37 @@ /* Kamke 2. 101 */ (pn_(101),ans:contrib_ode(eqn:x*'diff(y,x,2)+2*'diff(y,x,1)+a*x*y,y,x)); -false; +[y = %k2*%e^(sqrt(-a)*x)/x+%k1*%e^-(sqrt(-a)*x)/x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 102 */ (pn_(102),ans:contrib_ode(eqn:x*'diff(y,x,2)+2*'diff(y,x,1)+a*x^2*y,y,x)); -false; +[y = bessel_y(1/3,2*sqrt(a)*x^(3/2)/3)*%k2/sqrt(x) + +bessel_j(1/3,2*sqrt(a)*x^(3/2)/3)*%k1/sqrt(x)]; +method; +odelin; /* Kamke 2. 103 */ (pn_(103),ans:contrib_ode(eqn:x*'diff(y,x,2)-2*'diff(y,x,1)+a*y,y,x)); -false; +[y = bessel_y(-3,2*sqrt(a)*sqrt(x))*%k2*x^(3/2) + +bessel_j(-3,2*sqrt(a)*sqrt(x))*%k1*x^(3/2)]; +method; +odelin; /* Kamke 2. 104 */ (pn_(104),ans:contrib_ode(eqn:x*'diff(y,x,2)+v*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*bessel_y(v-1,2*sqrt(a)*sqrt(x))/x^(v/2-1/2) + +%k1*bessel_j(v-1,2*sqrt(a)*sqrt(x))/x^(v/2-1/2)]; +method; +odelin; /* Kamke 2. 105 */ (pn_(105),ans:contrib_ode(eqn:x*'diff(y,x,2)+a*'diff(y,x,1)+b*x*y,y,x)); -false; + [y = %k2*bessel_y(-(a-1)/2,-sqrt(b)*x)/x^(a/2-1/2) + +%k1*bessel_j(-(a-1)/2,-sqrt(b)*x)/x^(a/2-1/2)]; +method; +odelin; /* Kamke 2. 106 */ (pn_(106),ans:contrib_ode(eqn:x*'diff(y,x,2)+a*'diff(y,x,1)+b*x^a1*y,y,x)); @@ -35,11 +49,15 @@ /* Kamke 2. 107 */ (pn_(107),ans:contrib_ode(eqn:x*'diff(y,x,2)+(x+b)*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*kummer_u(a,b,-x)+%k1*kummer_m(a,b,-x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 108 */ (pn_(108),ans:contrib_ode(eqn:x*'diff(y,x,2)+(x+b+a)*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*kummer_u(a,b+a,-x)+%k1*kummer_m(a,b+a,-x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 109 */ (pn_(109),ans:contrib_ode(eqn:x*'diff(y,x,2)-x*'diff(y,x,1)-y-x*(x+1)*%e^x,y,x)); @@ -49,71 +67,97 @@ method; variationofparameters; -/* Kamke 2. 110 */ +/* Kamke 2. 110 */ assume(x>0,x+4*a>0); [x>0,x+4*a>0]; (pn_(110),ans:contrib_ode(eqn:x*'diff(y,x,2)-x*'diff(y,x,1)-a*y,y,x)); -[[]]; -method; -freeofx; +[y = %k2*kummer_u(-a,0,-x)*%e^x+%k1*kummer_m(-a,0,-x)*%e^x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; forget(x>0,x+4*a>0); [x>0,x+4*a>0]; -/* Kamke 2. 111 */ +/* Kamke 2. 111 */ (pn_(111),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x+1)*'diff(y,x,1)+y,y,x)); -[[]]; -method; -freeofx; +[y = %k2*%e^x+%k1*(x+1)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 112 */ -(pn_(112),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x+1)*'diff(y,x,1)-2*(x-1)*y,y,x)); -false; +(pn_(112),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x+1)*'diff(y,x,1)-2*(x-1)*y,y,x));\ +[y = %k2*%e^(2*x)+%k1*(3*x+1)*%e^-x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 113 */ assume(x^2+(4*a-2*b)*x+b^2>0); [x^2+(4*a-2*b)*x+b^2>0]; (pn_(113),ans:contrib_ode(eqn:x*'diff(y,x,2)+(b-x)*'diff(y,x,1)-a*y,y,x)); -[[]]; -method; -freeofx; +[y = %k2*kummer_u(b-a,b,-x)*%e^x+%k1*kummer_m(b-a,b,-x)*%e^x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; forget(x^2+(4*a-2*b)*x+b^2>0); [x^2+(4*a-2*b)*x+b^2>0]; /* Kamke 2. 114 */ (pn_(114),ans:contrib_ode(eqn:x*'diff(y,x,2)-2*(x-1)*'diff(y,x,1)-y,y,x)); -false; +[y = kummer_u(1/2,0,-2*x)*%k2*%e^(2*x)/x + +kummer_m(1/2,0,-2*x)*%k1*%e^(2*x)/x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 115 */ (pn_(115),ans:contrib_ode(eqn:x*'diff(y,x,2)-(3*x-2)*'diff(y,x,1)+(3-2*x)*y,y,x)); -false; +[y = kummer_u(6*sqrt(17)/17,0,-sqrt(17)*x)*%k2*%e^((sqrt(17)+3)*x/2)/x + +kummer_m(6*sqrt(17)/17,0,-sqrt(17)*x)*%k1*%e^((sqrt(17)+3)*x/2)/x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 116 */ -(pn_(116),ans:contrib_ode(eqn:x*'diff(y,x,2)+(a*x+n+b)*'diff(y,x,1)+a*n*y,y,x)); -false; +(pn_(116),ans:contrib_ode(eqn:x*'diff(y,x,2)+(a*x+n+b)*'diff(y,x,1)+a*n*y,y,x));[y = %k2*kummer_u(n,n+b,-a*x)+%k1*kummer_m(n,n+b,-a*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 117 */ (pn_(117),ans:contrib_ode(eqn:x*'diff(y,x,2)-(b+a)*(x+1)*'diff(y,x,1)+a*b*x*y,y,x)); -false; +[y = %k2*kummer_u(-b*(b+a)/(b-a),-b-a,-(b-a)*x)*%e^(b*x) + +%k1*kummer_m(-b*(b+a)/(b-a),-b-a,-(b-a)*x)*%e^(b*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 118 */ (pn_(118),ans:contrib_ode(eqn:x*'diff(y,x,2)+((b+a)*x+n+m)*'diff(y,x,1)+(a*b*x+a*n+b*m)*y,y,x)); -false; +[y = %k2*kummer_u(m,n+m,-(b-a)*x)*%e^-(a*x) + +%k1*kummer_m(m,n+m,-(b-a)*x)*%e^-(a*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; -/* Kamke 2. 119 */ +/* Kamke 2. 119 */ +/* CHECKME: Takes a long time (pn_(119),ans:contrib_ode(eqn:x*'diff(y,x,2)-2*(a*x+b)*'diff(y,x,1)+(a^2*x+2*a*b)*y,y,x)); false; +*/ /* Kamke 2. 120 */ (pn_(120),ans:contrib_ode(eqn:x*'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+(c*x+d)*y,y,x)); -false; +[y = %k2*kummer_u(-(sqrt(a^2-4*c)*(2*d-a*b)-4*b*c+a^2*b)/(2*(4*c-a^2)), + b,-sqrt(a^2-4*c)*x)*%e^((sqrt(a^2-4*c)-a)*x/2) + +%k1*kummer_m(-(sqrt(a^2-4*c)*(2*d-a*b)-4*b*c+a^2*b)/(2*(4*c-a^2)), + b,-sqrt(a^2-4*c)*x)*%e^((sqrt(a^2-4*c)-a)*x/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 121 */ (pn_(121),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x^2-x)*'diff(y,x,1)+(x-1)*y,y,x)); -false; +[y = %k2*x*'integrate(%e^(x^2/2-x)/x^2,x)+%k1*x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 122 */ (pn_(122),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x^2-x-2)*'diff(y,x,1)-x*(x+3)*y,y,x)); -false; +[y = %k2*%e^(x^2/2)*'integrate(%e^(-x^2/2-x)/x^2,x)+%k1*%e^(x^2/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 123 */ assume(b>0,b>a^2); @@ -129,7 +173,9 @@ /* Kamke 2. 124 */ (pn_(124),ans:contrib_ode(eqn:x*'diff(y,x,2)-2*(x^2-a)*'diff(y,x,1)+2*n*x*y,y,x)); -false; +[y = %k2*kummer_u(-n/2,(2*a+1)/2,x^2)+%k1*kummer_m(-n/2,(2*a+1)/2,x^2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 125 */ (pn_(125),ans:contrib_ode(eqn:x*'diff(y,x,2)+(4*x^2-1)*'diff(y,x,1)-4*x^3*y-4*x^5,y,x)); @@ -142,8 +188,10 @@ variationofparameters; /* Kamke 2. 126 */ +/* CHECKME: Takes a long time (pn_(126),ans:contrib_ode(eqn:x*'diff(y,x,2)+(2*a*x^3-1)*'diff(y,x,1)+(a^2*x^3+a)*y,y,x)); false; +*/ /* Kamke 2. 127 */ (pn_(127),ans:contrib_ode(eqn:x*'diff(y,x,2)+(2*a*x*log(x)+1)*'diff(y,x,1)+(a^2*x*log(x)^2+a*log(x)+a)*y,y,x)); @@ -155,7 +203,9 @@ /* Kamke 2. 129 */ (pn_(129),ans:contrib_ode(eqn:(x-3)*'diff(y,x,2)-(4*x-9)*'diff(y,x,1)+(3*x-6)*y,y,x)); -false; +[y = %k2*(4*x^3-42*x^2+150*x-183)*%e^(3*x)+%k1*%e^x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 130 */ (pn_(130),ans:contrib_ode(eqn:2*x*'diff(y,x,2)+'diff(y,x,1)+a*y,y,x)); @@ -167,25 +217,33 @@ /* Kamke 2. 131 */ (pn_(131),ans:contrib_ode(eqn:2*x*'diff(y,x,2)-(x-1)*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*kummer_u(a+1,3/2,-x/2)*sqrt(x)*%e^(x/2) + +%k1*kummer_m(a+1,3/2,-x/2)*sqrt(x)*%e^(x/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 132 */ assume(4*x^2+(-8*a-4)*x+1>0); [4*x^2+(-8*a-4)*x+1>0]; (pn_(132),ans:contrib_ode(eqn:2*x*'diff(y,x,2)-(2*x-1)*'diff(y,x,1)+a*y,y,x)); -[[]]; -method; -freeofx; +[y = %k2*kummer_u((a+2)/2,3/2,-x)*sqrt(x)*%e^x + +%k1*kummer_m((a+2)/2,3/2,-x)*sqrt(x)*%e^x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; forget(4*x^2+(-8*a-4)*x+1>0); [4*x^2+(-8*a-4)*x+1>0]; /* Kamke 2. 133 */ (pn_(133),ans:contrib_ode(eqn:(2*x-1)*'diff(y,x,2)-(3*x-4)*'diff(y,x,1)+(x-3)*y,y,x)); -false; +[y = %k2*%e^x*'integrate(%e^-(x/2)/(2*x-1)^(5/4),x)+%k1*%e^x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 134 */ (pn_(134),ans:contrib_ode(eqn:4*x*'diff(y,x,2)-(x+a)*y,y,x)); -false; +[y = %k2*kummer_u(-a/4,0,-x)*%e^(x/2)+%k1*kummer_m(-a/4,0,-x)*%e^(x/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 135 */ (pn_(135),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+2*'diff(y,x,1)-y,y,x)); @@ -197,27 +255,40 @@ /* Kamke 2. 136 */ (pn_(136),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+4*'diff(y,x,1)+(-x-2)*y,y,x)); -false; +[y = %k2*%e^(x/2)*'integrate(%e^-x/x,x)+%k1*%e^(x/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 137 */ (pn_(137),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+(-x-2)*y+l*y+4*y,y,x)); -false; +[y = %k2*kummer_u((l+2)/4,0,-x)*%e^(x/2) + +%k1*kummer_m((l+2)/4,0,-x)*%e^(x/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 138 */ (pn_(138),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+4*m*'diff(y,x,1)+(-x+4*n+2*m)*y,y,x)); -false; +[y = %k2*kummer_u(n+m,m,-x)*%e^(x/2)+%k1*kummer_m(n+m,m,-x)*%e^(x/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 139 */ (pn_(139),ans:contrib_ode(eqn:16*x*'diff(y,x,2)+8*'diff(y,x,1)+(-x-a)*y,y,x)); -false; +[y = %k2*kummer_u(-(a-6)/8,3/2,-x/2)*sqrt(x)*%e^(x/4) + +%k1*kummer_m(-(a-6)/8,3/2,-x/2)*sqrt(x)*%e^(x/4)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 140 */ (pn_(140),ans:contrib_ode(eqn:a*x*'diff(y,x,2)+b*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*bessel_y((b-a)/a,2*sqrt(c)*sqrt(x)/sqrt(a))/x^(b/(2*a)-1/2) + +%k1*bessel_j((b-a)/a,2*sqrt(c)*sqrt(x)/sqrt(a))/x^(b/(2*a)-1/2)]; /* Kamke 2. 141 */ (pn_(141),ans:contrib_ode(eqn:a*x*'diff(y,x,2)+(b*x+3*a)*'diff(y,x,1)+3*b*y,y,x)); -false; +[y = %k2*%e^-(b*x/a)*'integrate(%e^(b*x/a)/x^3,x)+%k1*%e^-(b*x/a)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 142 */ (pn_(142),ans:contrib_ode(eqn:5*(a*x+b)*'diff(y,x,2)+8*a*'diff(y,x,1)+c*(a*x+b)^(1/5)*y,y,x)); @@ -225,15 +296,32 @@ /* Kamke 2. 143 */ (pn_(143),ans:contrib_ode(eqn:2*a*x*'diff(y,x,2)+(b*x+a)*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*kummer_u(c/b,1/2,-b*x/(2*a))+%k1*kummer_m(c/b,1/2,-b*x/(2*a))]; +[method, ode_check(eqn,ans[1])]; +odelin,0]; /* Kamke 2. 144 */ (pn_(144),ans:contrib_ode(eqn:2*a*x*'diff(y,x,2)+(b*x+3*a)*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*kummer_u(c/b,3/2,-b*x/(2*a))+%k1*kummer_m(c/b,3/2,-b*x/(2*a))]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; -/* Kamke 2. 145 */ +/* Kamke 2. 145 */ (pn_(145),ans:contrib_ode(eqn:(a2*x+b2)*'diff(y,x,2)+(a1*x+b1)*'diff(y,x,1)+(a0*x+b0)*y,y,x)); -false; +[y = %k2*kummer_u((sqrt(a1^2-4*a0*a2) + *((2*a0*a2-a1^2)*b2+a1*a2*b1-2*a2^2*b0) + +(a1^3-4*a0*a1*a2)*b2+(4*a0*a2^2-a1^2*a2)*b1) + /(2*a2^2*(4*a0*a2-a1^2)),-(a1*b2-a2*b1)/a2^2, + -sqrt(a1^2-4*a0*a2)*(a2*x+b2)/a2^2) + *%e^((sqrt(a1^2-4*a0*a2)*(a2*x+b2)-a1*a2*x)/(2*a2^2)) + +%k1*kummer_m((sqrt(a1^2-4*a0*a2) + *((2*a0*a2-a1^2)*b2+a1*a2*b1-2*a2^2*b0) + +(a1^3-4*a0*a1*a2)*b2+(4*a0*a2^2-a1^2*a2)*b1) + /(2*a2^2*(4*a0*a2-a1^2)),-(a1*b2-a2*b1)/a2^2, + -sqrt(a1^2-4*a0*a2)*(a2*x+b2)/a2^2) + *%e^((sqrt(a1^2-4*a0*a2)*(a2*x+b2)-a1*a2*x)/(2*a2^2))]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 146 */ assume(x>0); @@ -273,27 +361,45 @@ /* Kamke 2. 149 */ (pn_(149),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x+b)*y,y,x)); -false; +[y = %k2*bessel_y(sqrt(1-4*b),2*sqrt(a)*sqrt(x))*sqrt(x) + +%k1*bessel_j(sqrt(1-4*b),2*sqrt(a)*sqrt(x))*sqrt(x)]; +method; +odelin; /* Kamke 2. 150 */ (pn_(150),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(x^2-2)*y,y,x)); -false; +[y = bessel_y(-3/2,-x)*%k2*sqrt(-x)+bessel_j(-3/2,-x)*%k1*sqrt(-x)]; +method; +odelin; /* Kamke 2. 151 */ (pn_(151),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-(a*x^2+2)*y,y,x)); -false; +[y = bessel_y(-3/2,-sqrt(-a)*x)*%k2*sqrt(-x) + +bessel_j(-3/2,-sqrt(-a)*x)*%k1*sqrt(-x)]; +method; +odelin; /* Kamke 2. 152 */ (pn_(152),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a^2*x^2-6)*y,y,x)); -false; +[y = bessel_y(5/2,-a*x)*%k2*sqrt(-x)+bessel_j(5/2,-a*x)*%k1*sqrt(-x)]; +method; +odelin; /* Kamke 2. 153 */ (pn_(153),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x^2-(v-1)*v)*y,y,x)); -false; +[y = %k2*bessel_y(-(2*v-1)/2,-sqrt(a)*x)*sqrt(-x) + +%k1*bessel_j(-(2*v-1)/2,-sqrt(a)*x)*sqrt(-x)]; +method; +odelin; /* Kamke 2. 154 */ (pn_(154),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x^2+b*x+c)*y,y,x)); -false; +[y = %k2*kummer_u((a*sqrt(1-4*c)-sqrt(-a)*b+a)/(2*a),sqrt(1-4*c)+1, + -2*sqrt(-a)*x)*(-x)^((sqrt(1-4*c)+1)/2)*%e^(sqrt(-a)*x) + +%k1*kummer_m((a*sqrt(1-4*c)-sqrt(-a)*b+a)/(2*a),sqrt(1-4*c)+1, + -2*sqrt(-a)*x)*(-x)^((sqrt(1-4*c)+1)/2)*%e^(sqrt(-a)*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 155 */ (pn_(155),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x^k-(b-1)*b)*y,y,x)); @@ -309,7 +415,9 @@ /* Kamke 2. 158 */ (pn_(158),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*'diff(y,x,1)+(-b^2*x^2-a*b)*y,y,x)); -false; +[y = %k2*%e^(b*x)*'integrate(%e^(a/x-2*b*x),x)+%k1*%e^(b*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 159 */ (pn_(159),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*'diff(y,x,1)-y-a*x^2,y,x)); @@ -333,7 +441,9 @@ /* Kamke 2. 161 */ (pn_(161),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*'diff(y,x,1)+(-x-a)*y,y,x)); -false; +[y=%k2*bessel_y(2*sqrt(a),2*sqrt(-x))+%k1*bessel_j(2*sqrt(a),2*sqrt(-x))]; +method; +odelin; /* Kamke 2. 162 */ declare(v,integer); @@ -357,7 +467,9 @@ /* Kamke 2. 164 */ (pn_(164),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*'diff(y,x,1)+(l*x^2-v^2)*y,y,x)); -false; +[y = %k2*bessel_y(v,-sqrt(l)*x)+%k1*bessel_j(v,-sqrt(l)*x)]; +method; +odelin; /* Kamke 2. 165 */ (pn_(165),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(x+a)*'diff(y,x,1)-y,y,x)); @@ -389,23 +501,41 @@ /* Kamke 2. 169 */ (pn_(169),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x*'diff(y,x,1)+(a*x-b^2)*y,y,x)); -false; +[y = %k2*bessel_y(sqrt(4*b^2+1),2*sqrt(a)*sqrt(x))/sqrt(x) + +%k1*bessel_j(sqrt(4*b^2+1),2*sqrt(a)*sqrt(x))/sqrt(x)]; +method; +odelin; /* Kamke 2. 170 */ (pn_(170),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x*'diff(y,x,1)+(a*x^2+b)*y,y,x)); -false; +[y = %k2*bessel_y(sqrt(1-4*b)/2,-sqrt(a)*x)*sqrt(-x)/x + +%k1*bessel_j(sqrt(1-4*b)/2,-sqrt(a)*x)*sqrt(-x)/x]; +method; +odelin; /* Kamke 2. 171 */ (pn_(171),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x*'diff(y,x,1)+(l*x^2+a*x-n*(n+1))*y,y,x)); -false; +[y = %k2*kummer_u(-(2*l*n+a*sqrt(-l))/(2*l),-2*n,-2*sqrt(-l)*x) + *%e^(sqrt(-l)*x)/((-x)^n*x) + +%k1*kummer_m(-(2*l*n+a*sqrt(-l))/(2*l),-2*n,-2*sqrt(-l)*x) + *%e^(sqrt(-l)*x)/((-x)^n*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; + /* Kamke 2. 172 */ (pn_(172),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*(x-1)*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*bessel_y(sqrt(1-4*a)/2,-%i/x)*%e^-(1/x)/sqrt(-x) + +%k1*bessel_j(sqrt(1-4*a)/2,-%i/x)*%e^-(1/x)/sqrt(-x)]; +method; +odelin; /* Kamke 2. 173 */ (pn_(173),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*(x+a)*'diff(y,x,1)-(b-1)*b*y,y,x)); -false; + [y = %k2*bessel_y(-(2*b-1)/2,-%i*a/x)*%e^(a/x)/sqrt(-x) + +%k1*bessel_j(-(2*b-1)/2,-%i*a/x)*%e^(a/x)/sqrt(-x)]; +method; +odelin; /* Kamke 2. 174 */ (pn_(174),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*'diff(y,x,1)+2*y-x^5*log(x),y,x)); @@ -425,7 +555,9 @@ /* Kamke 2. 176 */ (pn_(176),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*'diff(y,x,1)+(x^2+2)*y,y,x)); -false; +[y = %k2*x*%e^(%i*x)+%k1*x*%e^-(%i*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 177 */ (pn_(177),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(x^2+2)*'diff(y,x,1)-2*x*'diff(y,x,1)-x^2/cos(x),y,x)); @@ -441,7 +573,9 @@ /* Kamke 2. 179 */ (pn_(179),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*'diff(y,x,1)+(a^2*x^2+2)*y,y,x)); -false; +[y = %k2*x*%e^(%i*a*x)+%k1*x*%e^-(%i*a*x)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 180 */ (pn_(180),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+3*x*'diff(y,x,1)+(x^2-v^2+1)*y-f(x),y,x)); @@ -481,7 +615,10 @@ /* Kamke 2. 185 */ (pn_(185),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+5*x*'diff(y,x,1)+(4-2*x^3)*y,y,x)); -false; +[y = bessel_y(0,-2*sqrt(2)*sqrt(-x)*x/3)*%k2/x^2 + +bessel_j(0,-2*sqrt(2)*sqrt(-x)*x/3)*%k1/x^2]; +method; +odelin; /* Kamke 2. 186 */ (pn_(186),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-5*x*'diff(y,x,1)+8*y-x^3*sin(x),y,x)); @@ -505,7 +642,12 @@ /* Kamke 2. 188 */ (pn_(188),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*kummer_u(-(sqrt(-4*c+a^2-2*a+1)+a-3)/2,1-sqrt(-4*c+a^2-2*a+1), + -b/x)*x^((sqrt(-4*c+a^2-2*a+1)-a+1)/2)*%e^(b/x) + +%k1*kummer_m(-(sqrt(-4*c+a^2-2*a+1)+a-3)/2,1-sqrt(-4*c+a^2-2*a+1), + -b/x)*x^((sqrt(-4*c+a^2-2*a+1)-a+1)/2)*%e^(b/x)]; +method; +odelin; /* Kamke 2. 189 */ (pn_(189),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*x*'diff(y,x,1)+(b*x^m+c)*y,y,x)); @@ -513,45 +655,75 @@ /* Kamke 2. 190 */ (pn_(190),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x^2*'diff(y,x,1)+(a*x+b)*y,y,x)); -false; +[y = %k2*kummer_u((sqrt(1-4*b)+2*a+1)/2,sqrt(1-4*b)+1,-x) + *(-x)^((sqrt(1-4*b)+1)/2) + +%k1*kummer_m((sqrt(1-4*b)+2*a+1)/2,sqrt(1-4*b)+1,-x) + *(-x)^((sqrt(1-4*b)+1)/2)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 191 */ (pn_(191),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x^2*'diff(y,x,1)-2*y,y,x)); -false; +[y = bessel_y(-3/2,-%i*x/2)*%k2*sqrt(-x)*%e^-(x/2) + +bessel_j(-3/2,-%i*x/2)*%k1*sqrt(-x)*%e^-(x/2)]; +method; +odelin; /* Kamke 2. 192 */ (pn_(192),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(x^2-1)*'diff(y,x,1)-y,y,x)); -false; + [y = %k2*%e^-x*'integrate(%e^(x-1/x),x)+%k1*%e^-x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 193 */ (pn_(193),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x+1)*'diff(y,x,1)+(x-9)*y,y,x)); -false; +[y = kummer_u(-2,-5,-x)*%k2/x^3+kummer_m(-2,-5,-x)*%k1/x^3]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 194 */ (pn_(194),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x+1)*'diff(y,x,1)+(3*x-1)*y,y,x)); -false; +[y = kummer_u(4,3,-x)*%k2*x+kummer_m(4,3,-x)*%k1*x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 195 */ (pn_(195),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x+3)*'diff(y,x,1)-y,y,x)); -false; +[y = kummer_u(sqrt(2)-1,2*sqrt(2)+1,-x)*%k2*x^(sqrt(2)-1) + +kummer_m(sqrt(2)-1,2*sqrt(2)+1,-x)*%k1*x^(sqrt(2)-1)]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; + /* Kamke 2. 196 */ (pn_(196),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-(x-1)*x*'diff(y,x,1)+(x-1)*y,y,x)); -false; +[y = %k2*x*'integrate(%e^x/x^3,x)+%k1*x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 197 */ (pn_(197),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-(x^2-2*x)*'diff(y,x,1)+(-x-a)*y,y,x)); -false; +[y = %k2*bessel_y(sqrt(4*a+1)/2,-%i*x/2)*sqrt(-x)*%e^(x/2)/x + +%k1*bessel_j(sqrt(4*a+1)/2,-%i*x/2)*sqrt(-x)*%e^(x/2)/x]; +method; +odelin; /* Kamke 2. 198 */ (pn_(198),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-(x^2-2*x)*'diff(y,x,1)+(-3*x-2)*y,y,x)); -false; +[y = %k2*x*%e^x*'integrate(%e^-x/x^4,x)+%k1*x*%e^x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 199 */ (pn_(199),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-x*(x+4)*'diff(y,x,1)+4*y,y,x)); -false; +[y = %k2*x^4*%e^x*'integrate(%e^-x/x^4,x)+%k1*x^4*%e^x]; +[method, ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 200 */ (pn_(200),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x^2*'diff(y,x,1)-(v-1)*v*y,y,x)); -false; +[y = %k2*bessel_y(-(2*v-1)/2,-%i*x)*sqrt(-x)*%e^-x + +%k1*bessel_j(-(2*v-1)/2,-%i*x)*sqrt(-x)*%e^-x]; +method; +odelin; Index: rtestode_kamke_2_3.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_2_3.mac,v retrieving revision 1.3 retrieving revision 1.4 diff -u -d -r1.3 -r1.4 --- rtestode_kamke_2_3.mac 2 Sep 2006 12:22:36 -0000 1.3 +++ rtestode_kamke_2_3.mac 31 Jan 2007 12:44:12 -0000 1.4 @@ -11,19 +11,29 @@ /* Kamke 2. 201 */ (pn_(201),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(2*x+1)*'diff(y,x,1)-4*y,y,x)); -false; +[y = kummer_u(-2,-3,-2*x)*%k2/x^2+kummer_m(-2,-3,-2*x)*%k1/x^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 202 */ (pn_(202),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*(x+1)*'diff(y,x,1)+2*(x+1)*y,y,x)); -false; +[y = %k2*x*%e^(2*x)+%k1*x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 203 */ (pn_(203),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*x^2*'diff(y,x,1)-2*y,y,x)); -false; +[y = bessel_y(-3/2,-%i*a*x/2)*%k2*sqrt(-x)*%e^-(a*x/2) + +bessel_j(-3/2,-%i*a*x/2)*%k1*sqrt(-x)*%e^-(a*x/2)]; +method; +odelin; /* Kamke 2. 204 */ (pn_(204),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(2*b+a)*x^2*'diff(y,x,1)+(b*(b+a)*x^2-2)*y,y,x)); -false; +[y = bessel_y(-3/2,-%i*a*x/2)*%k2*sqrt(-x)*%e^-(b*x+a*x/2) + +bessel_j(-3/2,-%i*a*x/2)*%k1*sqrt(-x)*%e^-(b*x+a*x/2)]; +method; +odelin; /* Kamke 2. 205 */ (pn_(205),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*x^2*'diff(y,x,1)+f(x)*y,y,x)); @@ -31,27 +41,57 @@ /* Kamke 2. 206 */ (pn_(206),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(2*a*x+b)*'diff(y,x,1)+(c*x^2+a*b*x+d)*y,y,x)); -false; +[y=%k2*bessel_y(sqrt(-4*d+b^2-2*b+1)/2,-sqrt(c-a^2)*x)*%e^-(a*x)/x^(b/2-1/2) + +%k1*bessel_j(sqrt(-4*d+b^2-2*b+1)/2,-sqrt(c-a^2)*x)*%e^-(a*x)/x^(b/2-1/2)]; +method; +odelin; /* Kamke 2. 207 */ (pn_(207),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(a*x+b)*'diff(y,x,1)+(a1*x^2+b1*x+c_1)*y,y,x)); -false; +[y = %k2*kummer_u(((4*a1-a^2)*sqrt(-4*c_1+b^2-2*b+1) + +sqrt(a^2-4*a1)*(a*b-2*b1)+4*a1-a^2) + /(2*(4*a1-a^2)),sqrt(-4*c_1+b^2-2*b+1)+1, + -sqrt(a^2-4*a1)*x)*x^((sqrt(-4*c_1+b^2-2*b+1)-b+1)/2) + *%e^((sqrt(a^2-4*a1)-a)*x/2) + +%k1*kummer_m(((4*a1-a^2)*sqrt(-4*c_1+b^2-2*b+1) + +sqrt(a^2-4*a1)*(a*b-2*b1)+4*a1-a^2) + /(2*(4*a1-a^2)),sqrt(-4*c_1+b^2-2*b+1)+1, + -sqrt(a^2-4*a1)*x)*x^((sqrt(-4*c_1+b^2-2*b+1)-b+1)/2) + *%e^((sqrt(a^2-4*a1)-a)*x/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 208 */ (pn_(208),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x^3*'diff(y,x,1)+(x^2-2)*y,y,x)); -false; +[y = %k2*%e^-(x^2/2)*(sqrt(2)*sqrt(%pi)*%e^(x^2/2)*erf(x/sqrt(2))-2*x)/x + +%k1/x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 209 */ (pn_(209),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x^2+2)*'diff(y,x,1)+(x^2-2)*y,y,x)); -false; +[y = %k2*%e^-(x^2/2) + *(sqrt(2)*sqrt(%pi)*%i*erf(%i*x/sqrt(2))+2*x*%e^(x^2/2))/x^2 + +%k1*%e^-(x^2/2)/x^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 210 */ (pn_(210),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*(x^2-a)*'diff(y,x,1)+(2*n*x^2+a*((-1)^n-1))*y,y,x)); -false; +[y = %k2*kummer_u((sqrt(-4*a*(-1)^n+4*a^2+1)-2*n-2*a+1)/4, + (sqrt(-4*a*(-1)^n+4*a^2+1)+2)/2,x^2) + *x^((sqrt(-4*a*(-1)^n+4*a^2+1)-2*a+1)/2) + +%k1*kummer_m((sqrt(-4*a*(-1)^n+4*a^2+1)-2*n-2*a+1)/4, + (sqrt(-4*a*(-1)^n+4*a^2+1)+2)/2,x^2) + *x^((sqrt(-4*a*(-1)^n+4*a^2+1)-2*a+1)/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 211 */ (pn_(211),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+4*x^3*'diff(y,x,1)+(4*x^4+2*x^2+1)*y,y,x)); -false; +[y = %k2*x^((sqrt(-3)+1)/2)*%e^-x^2+%k1*%e^-x^2/x^(sqrt(-3)/2-1/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 212 */ (pn_(212),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(a*x^2+b)*'diff(y,x,1)+f(x)*y,y,x)); @@ -59,11 +99,21 @@ /* Kamke 2. 213 */ (pn_(213),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x^3+1)*'diff(y,x,1)-y,y,x)); -false; + [y = kummer_u(2/3,1/3,x^3/3)*%k2*%e^-(x^3/3)/x + +kummer_m(2/3,1/3,x^3/3)*%k1*%e^-(x^3/3)/x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 214 */ (pn_(214),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(-x^4+(2*n+2*a+1)*x^2+a*(-1)^n-a^2)*y,y,x)); -false; +[y = %k2*kummer_u((sqrt(-4*a*(-1)^n+4*a^2+1)-2*n-2*a+1)/4, + (sqrt(-4*a*(-1)^n+4*a^2+1)+2)/2,x^2) + *x^((sqrt(-4*a*(-1)^n+4*a^2+1)+1)/2)*%e^-(x^2/2) + +%k1*kummer_m((sqrt(-4*a*(-1)^n+4*a^2+1)-2*n-2*a+1)/4, + (sqrt(-4*a*(-1)^n+4*a^2+1)+2)/2,x^2) + *x^((sqrt(-4*a*(-1)^n+4*a^2+1)+1)/2)*%e^-(x^2/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 215 */ (pn_(215),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(a*x^n+b)*'diff(y,x,1)+(a1*x^(2*n)+b1*x^n+c_1)*y,y,x)); @@ -123,19 +173,34 @@ /* Kamke 2. 225 */ (pn_(225),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,2)-x*'diff(y,x,1)+y,y,x)); -false; +[y = gauss_b(2,2,5/2,-(%i*x-1)/2)*%k2*(-%i*x-1)^(1/4)*(1-%i*x)^(1/4) + *(x^2+1)^(5/4) + +gauss_a(2,2,5/2,-(%i*x-1)/2)*%k1*(-%i*x-1)^(1/4)*(1-%i*x)^(1/4) + *(x^2+1)^(5/4)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 226 */ (pn_(226),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,2)+2*x*'diff(y,x,1)-(v-1)*v*y,y,x)); -false; +[y = %k2*gauss_b(1-v,v,1,-(%i*x-1)/2)*sqrt(-%i*x-1)*sqrt(1-%i*x)/sqrt(x^2+1) + +%k1*gauss_a(1-v,v,1,-(%i*x-1)/2)*sqrt(-%i*x-1)*sqrt(1-%i*x)/sqrt(x^2+1)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 227 */ (pn_(227),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,2)-2*x*'diff(y,x,1)+2*y,y,x)); -false; +[y = %k2*(x^2+1)*%e^(2*%i*atan(x))+%k1*(x^2+1)*%e^-(2*%i*atan(x))]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 228 */ (pn_(228),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,2)+3*x*'diff(y,x,1)+a*y,y,x)); -false; +[y = %k2*gauss_b(sqrt(1-a),-sqrt(1-a),1/2,-(%i*x-1)/2)*(-%i*x-1)^(1/4) + *(1-%i*x)^(1/4)/(x^2+1)^(3/4) + +%k1*gauss_a(sqrt(1-a),-sqrt(1-a),1/2,-(%i*x-1)/2)*(-%i*x-1)^(1/4) + *(1-%i*x)^(1/4)/(x^2+1)^(3/4)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 229 */ (pn_(229),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,2)+4*x*'diff(y,x,1)+2*y-2*cos(x)+2*x,y,x)); @@ -155,7 +220,9 @@ /* Kamke 2. 231 */ (pn_(231),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-v*(v+1)*y,y,x)); -false; +[y = %k2*gauss_b(-v-1,v,0,-(x-1)/2)+%k1*gauss_a(-v-1,v,0,-(x-1)/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 232 */ (pn_(232),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-n*(n+1)*y+Diff(LegendreP(n,x),x),y,x)); @@ -203,31 +270,55 @@ /* Kamke 2. 239 */ (pn_(239),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+2*x*'diff(y,x,1)-l*y,y,x)); -false; + [y = %k2*gauss_b((sqrt(4*l+1)+1)/2,-(sqrt(4*l+1)-1)/2,1,-(x-1)/2) + +%k1*gauss_a((sqrt(4*l+1)+1)/2,-(sqrt(4*l+1)-1)/2,1,-(x-1)/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 240 */ (pn_(240),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+2*x*'diff(y,x,1)-v*(v+1)*y,y,x)); -false; +[y = %k2*gauss_b(-v,v+1,1,-(x-1)/2)+%k1*gauss_a(-v,v+1,1,-(x-1)/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 241 */ (pn_(241),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-2*x*'diff(y,x,1)+(-v-2)*(v-1)*y,y,x)); -false; +[y = %k2*gauss_b(-v,v+1,3,-(x-1)/2)*(x-1)^2 + +%k1*gauss_a(-v,v+1,3,-(x-1)/2)*(x-1)^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 242 */ (pn_(242),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-(3*x+1)*'diff(y,x,1)+(x-x^2)*y,y,x)); -false; + [y = %k2*%e^-x + *(x^3*'integrate((8*x-7)*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1),x) + +3*x^2*'integrate((8*x-7)*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1),x) + +3*x*'integrate((8*x-7)*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1),x) + +'integrate((8*x-7)*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1),x) + -2*x^2*%e^(2*x)+3*x*%e^(2*x))/(x+1) + +%k1*(x+1)^2*%e^-x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 243 */ (pn_(243),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+4*x*'diff(y,x,1)+(x^2+1)*y,y,x)); -false; +[y = %k2*%e^(%i*x)/((x-1)*(x+1))+%k1*%e^-(%i*x)/((x-1)*(x+1))]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 244 */ (pn_(244),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+2*(n+1)*x*'diff(y,x,1)+(-v-n-1)*(v-n)*y,y,x)); -false; + [y = %k2*gauss_b(-v,v+1,n+1,-(x-1)/2)/(x+1)^n + +%k1*gauss_a(-v,v+1,n+1,-(x-1)/2)/(x+1)^n]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 245 */ (pn_(245),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-2*(n-1)*x*'diff(y,x,1)+(-v+n-1)*(v+n)*y,y,x)); -false; +[y = %k2*gauss_b(-v,v+1,n+1,-(x-1)/2)*(x-1)^n + +%k1*gauss_a(-v,v+1,n+1,-(x-1)/2)*(x-1)^n]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 246 */ (pn_(246),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-2*(v-1)*x*'diff(y,x,1)-2*v*y,y,x)); @@ -239,27 +330,43 @@ /* Kamke 2. 247 */ (pn_(247),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+2*a*x*'diff(y,x,1)+(a-1)*a*y,y,x)); -false; + [y = %k2*(x+1)^(1-a)*'integrate((x+1)^a/((x-1)^a*(x^2+2*x+1)),x) + +%k1*(x+1)^(1-a)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 248 */ +/* Take a long time (pn_(248),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+a*x*'diff(y,x,1)+(b*x^2+c*x+d)*y,y,x)); false; +*/ /* Kamke 2. 249 */ +/* Take a long time (pn_(249),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+c*y,y,x)); false; +*/ /* Kamke 2. 250 */ (pn_(250),ans:contrib_ode(eqn:(x^2-a^2)*'diff(y,x,2)+8*x*'diff(y,x,1)+12*y,y,x)); -false; +[y = %k2*(3*x^2+a^2)/((x-a)^3*(x+a)^3)+%k1/(x+a)^3]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 251 */ (pn_(251),ans:contrib_ode(eqn:x*(x+1)*'diff(y,x,2)-(x-1)*'diff(y,x,1)+y,y,x)); -false; +[y = gauss_b(2,2,1,-x)*%k2*(x+1)^3+gauss_a(2,2,1,-x)*%k1*(x+1)^3]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 252 */ (pn_(252),ans:contrib_ode(eqn:x*(x+1)*'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*gauss_b((sqrt(-4*c+a^2-2*a+1)+a-1)/2, + -(sqrt(-4*c+a^2-2*a+1)-a+1)/2,b,-x) + +%k1*gauss_a((sqrt(-4*c+a^2-2*a+1)+a-1)/2, + -(sqrt(-4*c+a^2-2*a+1)-a+1)/2,b,-x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 253 */ (pn_(253),ans:contrib_ode(eqn:x*(x+1)*'diff(y,x,2)+(3*x+2)*'diff(y,x,1)+y,y,x)); @@ -271,7 +378,16 @@ /* Kamke 2. 254 */ (pn_(254),ans:contrib_ode(eqn:(x^2+x-2)*'diff(y,x,2)+(x^2-x)*'diff(y,x,1)+(-6*x^2-7*x)*y,y,x)); -false; + [y = %k2*%e^-(3*x) + *(2*x^2*%e^(5*x) + *'integrate((7*x-12)*%e^-(5*x)/(x^3-3*x^2+3*x-1),x) + -4*x*%e^(5*x) + *'integrate((7*x-12)*%e^-(5*x)/(x^3-3*x^2+3*x-1),x) + +2*%e^(5*x)*'integrate((7*x-12)*%e^-(5*x)/(x^3-3*x^2+3*x-1),x) + -x^2-4*x)/(x-1) + +%k1*(x-1)*%e^(2*x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 255 */ (pn_(255),ans:contrib_ode(eqn:(x-1)*x*'diff(y,x,2)+a*'diff(y,x,1)-2*y,y,x)); @@ -283,7 +399,9 @@ /* Kamke 2. 256 */ (pn_(256),ans:contrib_ode(eqn:(x-1)*x*'diff(y,x,2)+(2*x-1)*'diff(y,x,1)-v*(v+1)*y,y,x)); -false; +[y = %k2*gauss_b(-v,v+1,1,x)+%k1*gauss_a(-v,v+1,1,x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 257 */ (pn_(257),ans:contrib_ode(eqn:(x-1)*x*'diff(y,x,2)+((a+1)*x+b)*'diff(y,x,1),y,x)); @@ -295,11 +413,19 @@ /* Kamke 2. 258 */ (pn_(258),ans:contrib_ode(eqn:(x-1)*x*'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*gauss_b((sqrt(-4*c+a^2-2*a+1)+a-1)/2, + -(sqrt(-4*c+a^2-2*a+1)-a+1)/2,-b,x) + +%k1*gauss_a((sqrt(-4*c+a^2-2*a+1)+a-1)/2, + -(sqrt(-4*c+a^2-2*a+1)-a+1)/2,-b,x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 259 */ (pn_(259),ans:contrib_ode(eqn:(x-1)*x*'diff(y,x,2)+((a+1)*x+b)*'diff(y,x,1)-l*y,y,x)); -false; +[y = %k2*gauss_b((sqrt(4*l+a^2)+a)/2,-(sqrt(4*l+a^2)-a)/2,-b,x) + +%k1*gauss_a((sqrt(4*l+a^2)+a)/2,-(sqrt(4*l+a^2)-a)/2,-b,x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 260 */ (pn_(260),ans:contrib_ode(eqn:(x-1)*x*'diff(y,x,2)+((b1+a1+1)*x-d1)*'diff(y,x,1)+a1*b1*d1,y,x)); @@ -315,7 +441,10 @@ /* Kamke 2. 262 */ (pn_(262),ans:contrib_ode(eqn:(x+1)^2*'diff(y,x,2)+(x^2+x-1)*'diff(y,x,1)+(-x-2)*y,y,x)); -false; +[y = %k2*%e^-x*'integrate((x+1)*%e^(x^2/(x+1)+x/(x+1)-1/(x+1)),x) + +%k1*%e^-x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 263 CHECKME - seems OK */ (pn_(263),ans:contrib_ode(eqn:x*(x+3)*'diff(y,x,2)+(3*x-1)*'diff(y,x,1)+y+(-20*x-30)*(x^2+3*x)^(7/3),y,x)); @@ -327,7 +456,9 @@ /* Kamke 2. 264 */ (pn_(264),ans:contrib_ode(eqn:(x^2+3*x+4)*'diff(y,x,2)+(x^2+x+1)*'diff(y,x,1)+(-2*x-3)*y,y,x)); -false; +[y = %k2*%e^-x+%k1*(x^2+x+3)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 265 */ (pn_(265),ans:contrib_ode(eqn:(x-2)*(x-1)*'diff(y,x,2)-(2*x-3)*'diff(y,x,1)+y,y,x)); @@ -355,11 +486,30 @@ /* Kamke 2. 269 */ (pn_(269),ans:contrib_ode(eqn:2*(x-1)*x*'diff(y,x,2)+((2*v+5)*x-2*v-3)*'diff(y,x,1)+(v+1)*y,y,x)); -false; +[y = %k2*gauss_b(-v,1/2,-(2*v-1)/2,x)/x^(v+1/2) + +%k1*gauss_a(-v,1/2,-(2*v-1)/2,x)/x^(v+1/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 270 */ (pn_(270),ans:contrib_ode(eqn:(2*x^2+6*x+4)*'diff(y,x,2)+(10*x^2+21*x+8)*'diff(y,x,1)+(12*x^2+17*x+8)*y,y,x)); -false; +[y = %k2*%e^-(3*x) + *(x^5*sqrt(x+1)*'integrate((43*x+36)*%e^x + /(sqrt(x+1)*(x^6+12*x^5+60*x^4+160*x^3+240*x^2+192*x+64)),x) + +10*x^4*sqrt(x+1)*'integrate((43*x+36)*%e^x + /(sqrt(x+1)*(x^6+12*x^5+60*x^4+160*x^3+240*x^2+192*x+64)),x) + +40*x^3*sqrt(x+1)*'integrate((43*x+36)*%e^x + /(sqrt(x+1)*(x^6+12*x^5+60*x^4+160*x^3+240*x^2+192*x+64)),x) + +80*x^2*sqrt(x+1)*'integrate((43*x+36)*%e^x + /(sqrt(x+1)*(x^6+12*x^5+60*x^4+160*x^3+240*x^2+192*x+64)),x) + +80*x*sqrt(x+1)*'integrate((43*x+36)*%e^x + /(sqrt(x+1)*(x^6+12*x^5+60*x^4+160*x^3+240*x^2+192*x+64)),x) + +32*sqrt(x+1)*'integrate((43*x+36)*%e^x + /(sqrt(x+1)*(x^6+12*x^5+60*x^4+160*x^3+240*x^2+192*x+64)),x) + +4*x^2*%e^x+22*x*%e^x+18*%e^x)/(sqrt(x+1)*(x+2)) + +%k1*(x+2)^4*%e^-(3*x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 271 */ (pn_(271),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+y,y,x)); @@ -371,15 +521,22 @@ /* Kamke 2. 272 */ (pn_(272),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+(4*a^2*x^2+1)*y,y,x)); -false; +[y = bessel_y(0,-a*x)*%k2*sqrt(-x)+bessel_j(0,-a*x)*%k1*sqrt(-x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 273 */ (pn_(273),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)-(x^2-4*k*x+4*m^2-1)*y,y,x)); -false; +[y = %k2*kummer_u(-(2*m-2*k-1)/2,1-2*m,-x)*%e^(x/2)/(-x)^(m-1/2) + +%k1*kummer_m(-(2*m-2*k-1)/2,1-2*m,-x)*%e^(x/2)/(-x)^(m-1/2)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 274 */ (pn_(274),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+4*x*'diff(y,x,1)+(x-v^2)*y,y,x)); -false; +[y = %k2*bessel_y(v,sqrt(x))+%k1*bessel_j(v,sqrt(x))]; +method; +odelin; /* Kamke 2. 275 */ (pn_(275),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+4*x*'diff(y,x,1)+(-x^2+2*(-m+2*l+1)*x-m^2+1)*y,y,x)); @@ -391,7 +548,9 @@ /* Kamke 2. 277 */ (pn_(277),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+4*x*'diff(y,x,1)+(-a*x^2-1)*y,y,x)); -false; +[y = %k2*%e^(sqrt(a)*x/2)/sqrt(x)+%k1*%e^-(sqrt(a)*x/2)/sqrt(x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 278 */ (pn_(278),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+4*x*'diff(y,x,1)+f(x)*y,y,x)); @@ -407,15 +566,21 @@ /* Kamke 2. 280 */ (pn_(280),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+8*x*'diff(y,x,1)+(-4*x^2-12*x-3)*y,y,x)); -false; +[y = %k2*sqrt(x)*%e^x*'integrate(%e^-(2*x)/x^3,x)+%k1*sqrt(x)*%e^x]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 281 */ (pn_(281),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)-4*x*(2*x-1)*'diff(y,x,1)+(4*x^2-4*x-1)*y,y,x)); -false; +[y = %k2*sqrt(x)*%e^x+%k1*%e^x/sqrt(x)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 282 */ (pn_(282),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+4*x^3*'diff(y,x,1)+(x^2-4)*(x^2+6)*y,y,x)); -false; +[y = %k2*x^3*%e^-(x^2/4)+%k1*%e^-(x^2/4)/x^2]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 283 */ (pn_(283),ans:contrib_ode(eqn:4*x^2*'diff(y,x,2)+4*x^2*log(x)*'diff(y,x,1)+(x^2*log(x)^2+2*x-8)*y-4*x^2*%e^(x/2)/sqrt(x^x),y,x)); @@ -431,7 +596,10 @@ /* Kamke 2. 285 */ (pn_(285),ans:contrib_ode(eqn:x*(4*x-1)*'diff(y,x,2)+((4*a+2)*x-a)*'diff(y,x,1)+(a-1)*a*y,y,x)); -false; +[y = %k2*gauss_b(a/2,(a+1)/2,a,4*x)*sqrt(4*x-1) + +%k1*gauss_a(a/2,(a+1)/2,a,4*x)*sqrt(4*x-1)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 286 */ (pn_(286),ans:contrib_ode(eqn:(3*x-1)^2*'diff(y,x,2)+3*(3*x-1)*'diff(y,x,1)-9*y-log(3*x-1)^2,y,x)); @@ -443,15 +611,23 @@ /* Kamke 2. 287 */ (pn_(287),ans:contrib_ode(eqn:9*(x-1)*x*'diff(y,x,2)+3*(2*x-1)*'diff(y,x,1)-20*y,y,x)); -false; +[y = gauss_b(-1,2,5/3,x)*%k2*x^(2/3)+gauss_a(-1,2,5/3,x)*%k1*x^(2/3)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 288 */ (pn_(288),ans:contrib_ode(eqn:16*x^2*'diff(y,x,2)+(4*x+3)*y,y,x)); -false; +[y = bessel_y(-1/2,sqrt(x))*%k2*sqrt(x) + +bessel_j(-1/2,sqrt(x))*%k1*sqrt(x)]; +method; +odelin; /* Kamke 2. 289 */ (pn_(289),ans:contrib_ode(eqn:16*x^2*'diff(y,x,2)+32*x*'diff(y,x,1)+(-4*x-5)*y,y,x)); -false; +[y = bessel_y(-3/2,sqrt(-x))*%k2*sqrt(-x)/x + +bessel_j(-3/2,sqrt(-x))*%k1*sqrt(-x)/x]; +method; +odelin /* Kamke 2. 290 */ assume(x>0); @@ -475,11 +651,17 @@ /* Kamke 2. 293 */ (pn_(293),ans:contrib_ode(eqn:144*(x-1)*x*'diff(y,x,2)+(120*x-48)*'diff(y,x,1)+y,y,x)); -false; +[y = gauss_b(7/12,7/12,5/3,x)*%k2*x^(2/3) + +gauss_a(7/12,7/12,5/3,x)*%k1*x^(2/3)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 294 */ (pn_(294),ans:contrib_ode(eqn:144*(x-1)*x*'diff(y,x,2)+(168*x-96)*'diff(y,x,1)+y,y,x)); -false; +[y = gauss_b(5/12,5/12,4/3,x)*%k2*x^(1/3) + +gauss_a(5/12,5/12,4/3,x)*%k1*x^(1/3)]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 295 */ (pn_(295),ans:contrib_ode(eqn:a*x^2*'diff(y,x,2)+b*x*'diff(y,x,1)+(c*x^2+d*x+f)*y,y,x)); @@ -487,7 +669,24 @@ /* Kamke 2. 296 */ (pn_(296),ans:contrib_ode(eqn:a2*x^2*'diff(y,x,2)+(a1*x^2+b1*x)*'diff(y,x,1)+(a0*x^2+b0*x+c_0)*y,y,x)); -false; +[y = %k2*kummer_u(((4*a0*a2-a1^2)*sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2) + +sqrt(a1^2-4*a0*a2)*(a1*b1-2*a2*b0)+4*a0*a2^2 + -a1^2*a2) + /(2*a2*(4*a0*a2-a1^2)), + (sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2)+a2)/a2, + -sqrt(a1^2-4*a0*a2)*x/a2) + *x^((sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2)-b1+a2)/(2*a2)) + *%e^((sqrt(a1^2-4*a0*a2)-a1)*x/(2*a2)) + +%k1*kummer_m(((4*a0*a2-a1^2)*sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2) + +sqrt(a1^2-4*a0*a2)*(a1*b1-2*a2*b0)+4*a0*a2^2 + -a1^2*a2) + /(2*a2*(4*a0*a2-a1^2)), + (sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2)+a2)/a2, + -sqrt(a1^2-4*a0*a2)*x/a2) + *x^((sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2)-b1+a2)/(2*a2)) + *%e^((sqrt(a1^2-4*a0*a2)-a1)*x/(2*a2))]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 297 */ assume(a>0); @@ -503,7 +702,18 @@ /* Kamke 2. 298 */ (pn_(298),ans:contrib_ode(eqn:(a*x^2+1)*'diff(y,x,2)+b*x*'diff(y,x,1)+c*y,y,x)); -false; +[y = %k2*gauss_b((sqrt(-4*a*c+b^2-2*a*b+a^2)+a)/(2*a), + -(sqrt(-4*a*c+b^2-2*a*b+a^2)-a)/(2*a),b/(2*a), + -(sqrt(-a)*x-1)/2)*(1-sqrt(-a)*x)^(b/(4*a)) + *(sqrt(-a)*x+1) + /((-sqrt(-a)*x-1)^(b/(4*a))*(a*x^2+1)^(b/(4*a))) + +%k1*gauss_a((sqrt(-4*a*c+b^2-2*a*b+a^2)+a)/(2*a), + -(sqrt(-4*a*c+b^2-2*a*b+a^2)-a)/(2*a),b/(2*a), + -(sqrt(-a)*x-1)/2)*(1-sqrt(-a)*x)^(b/(4*a)) + *(sqrt(-a)*x+1) + /((-sqrt(-a)*x-1)^(b/(4*a))*(a*x^2+1)^(b/(4*a)))]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; /* Kamke 2. 299 */ (pn_(299),ans:contrib_ode(eqn:(a^2*x^2-1)*'diff(y,x,2)+2*a^2*x*'diff(y,x,1),y,x)); @@ -515,5 +725,7 @@ /* Kamke 2. 300 */ (pn_(300),ans:contrib_ode(eqn:(a^2*x^2-1)*'diff(y,x,2)+2*a^2*x*'diff(y,x,1)-2*a^2*y,y,x)); -false; +[y = gauss_b(-1,2,1,-(a*x-1)/2)*%k2+gauss_a(-1,2,1,-(a*x-1)/2)*%k1]; +[method,ode_check(eqn,ans[1])]; +[odelin,0]; ```

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