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[Maxima-commits] CVS: maxima/share/contrib/diffequations/tests rtestode_kamke_1_1.mac, 1.21, 1.22 rtestode_kamke_1_2.mac, 1.18, 1.19 rtestode_kamke_1_3.mac, 1.5, 1.6 rtestode_kamke_2_1.mac, 1.6, 1.7 rtestode_kamke_2_2.mac, 1.11, 1.12 rtestode_kamke_2_3.mac, 1.11, 1.12 rtestode_kamke_2_4.mac, 1.11, 1.12 rtestode_murphy_1_1.mac, 1.6, 1.7 rtestode_murphy_1_2.mac, 1.10, 1.11 rtestode_murphy_1_3.mac, 1.6, 1.7 rtestode_odelin.mac, 1.5, 1.6 testsuite.lisp, 1.16, 1.17
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From: David B. <bil...@us...> - 2007-12-26 03:43:03
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Update of /cvsroot/maxima/maxima/share/contrib/diffequations/tests In directory sc8-pr-cvs16.sourceforge.net:/tmp/cvs-serv27176/tests Modified Files: rtestode_kamke_1_1.mac rtestode_kamke_1_2.mac rtestode_kamke_1_3.mac rtestode_kamke_2_1.mac rtestode_kamke_2_2.mac rtestode_kamke_2_3.mac rtestode_kamke_2_4.mac rtestode_murphy_1_1.mac rtestode_murphy_1_2.mac rtestode_murphy_1_3.mac rtestode_odelin.mac testsuite.lisp Log Message: o Resolve testsuite issues by: - replacing $ratsimp with sratsimp in odelin.lisp and lazysolver.lisp - inserting ($expand ... 0) into function $odelin in odelin.lisp These changes recommended by Barton Willis in email of 2007-12-09 o Many changes to testsuite results: - a number of odelin tests now pass as due to above correction. - many odelin tests changed due to use of expand. o Update expected failures in testsuite.lisp. Index: rtestode_kamke_1_1.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_1_1.mac,v retrieving revision 1.21 retrieving revision 1.22 diff -u -d -r1.21 -r1.22 --- rtestode_kamke_1_1.mac 1 Dec 2007 21:55:14 -0000 1.21 +++ rtestode_kamke_1_1.mac 26 Dec 2007 03:42:56 -0000 1.22 @@ -160,9 +160,7 @@ /* 15 This is a Riccati eqn */ (pn_(15),ans:contrib_ode(eqn:'diff(y,x) + y^2 - 2*x^2*y + x^4 -2*x-1,y,x)); -[y = (x^2*(%c*%e^(x*(x^2+3)/3)+%e^(x*(x^2-3)/3)) - +%c*%e^(x*(x^2+3)/3)-%e^(x*(x^2-3)/3)) - /(%c*%e^(x*(x^2+3)/3)+%e^(x*(x^2-3)/3))]; + [y = (x^2*(%c*%e^(2*x)+1)+%c*%e^(2*x)-1)/(%c*%e^(2*x)+1)]; [method,ode_check(eqn,ans[1])]; [riccati,0]; @@ -182,11 +180,11 @@ /* 18 */ (pn_(18),ans:contrib_ode(eqn:'diff(y,x)-y^2-x*y-x+1,y,x)); -[y = -%e^-(2*x)*(2*%i*%e^(x^2/2+2) - +(sqrt(2)*sqrt(%pi)*erf(%i*(x-2)/sqrt(2)) - +sqrt(2)*%e^2*sqrt(%pi)*%c) - *%e^(2*x)) - /(sqrt(2)*sqrt(%pi)*erf(%i*(x-2)/sqrt(2))+sqrt(2)*%e^2*sqrt(%pi)*%c)]; +[y = -(%e^-(2*x)*(%e^(2*x)*(sqrt(2)*sqrt(%pi)*%c + *erf((%i*x-2*%i)/sqrt(2)) + +sqrt(2)*%e^2*sqrt(%pi)) + +2*%i*%c*%e^(x^2/2+2)) + /(sqrt(2)*sqrt(%pi)*%c*erf((%i*x-2*%i)/sqrt(2))+sqrt(2)*%e^2*sqrt(%pi)))]; [method,ode_check(eqn,ans[1])]; [riccati,0]; Index: rtestode_kamke_1_2.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_1_2.mac,v retrieving revision 1.18 retrieving revision 1.19 diff -u -d -r1.18 -r1.19 --- rtestode_kamke_1_2.mac 1 Dec 2007 21:55:14 -0000 1.18 +++ rtestode_kamke_1_2.mac 26 Dec 2007 03:42:56 -0000 1.19 @@ -394,10 +394,10 @@ /* 155 */ (pn_(155),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,1)+y^2-2*x*y+1,y,x)); -[y = (x^2*((x+1)*(%c*log(x+1)-%c*log(x-1)+1) - +(x-1)*(%c*log(x+1)-%c*log(x-1)+1)) - +(x+1)*(-%c*log(x+1)+%c*log(x-1)-4*%c*(x-1)-1) - +(x-1)*(-%c*log(x+1)+%c*log(x-1)-1)) +[y = (x^2*((x-1)*(%c*log(x+1)-%c*log(x-1)+2*%c+1) + +(x+1)*(%c*log(x+1)-%c*log(x-1)-2*%c+1)) + +(x+1)*(-%c*log(x+1)+%c*log(x-1)+2*%c-1) + +(x-1)*(-%c*log(x+1)+%c*log(x-1)-2*%c-1)) /((x-1)*(x+1)*(2*%c*log(x+1)-2*%c*log(x-1)+2))]; [method,ode_check(eqn,ans[1])]; [riccati,0]; @@ -495,27 +495,15 @@ /* 168 */ (pn_(168),ans:contrib_ode(eqn:3*(x^2-4)*'diff(y,x,1)+y^2-x*y-3,y,x)); -[y = -((x^2*(((3*dgauss_b(1/2,7/6,4/3,-((x-2)/4))*%c - +3*dgauss_a(1/2,7/6,4/3,-((x-2)/4))) - *(x-2) - -3*gauss_b(1/2,7/6,4/3,-((x-2)/4))*%c - -3*gauss_a(1/2,7/6,4/3,-((x-2)/4))) - *(x+2) - +(-3*gauss_b(1/2,7/6,4/3,-((x-2)/4))*%c - -3*gauss_a(1/2,7/6,4/3,-((x-2)/4))) - *(x-2)) - +((-12*dgauss_b(1/2,7/6,4/3,-((x-2)/4))*%c - -12*dgauss_a(1/2,7/6,4/3,-((x-2)/4))) - *(x-2) - +12*gauss_b(1/2,7/6,4/3,-((x-2)/4))*%c - +12*gauss_a(1/2,7/6,4/3,-((x-2)/4))) - *(x+2) - +(12*gauss_b(1/2,7/6,4/3,-((x-2)/4))*%c - +12*gauss_a(1/2,7/6,4/3,-((x-2)/4))) - *(x-2)) - /((4*gauss_b(1/2,7/6,4/3,-((x-2)/4))*%c - +4*gauss_a(1/2,7/6,4/3,-((x-2)/4))) - *(x-2)*(x+2)))]; +[y = -( ((3*dgauss_b(1/2,7/6,4/3,-((x-2)/4))*%c + +3*dgauss_a(1/2,7/6,4/3,-((x-2)/4))) + *x^2 + +(-6*gauss_b(1/2,7/6,4/3,-((x-2)/4))*%c + -6*gauss_a(1/2,7/6,4/3,-((x-2)/4))) + *x-12*dgauss_b(1/2,7/6,4/3,-((x-2)/4))*%c + -12*dgauss_a(1/2,7/6,4/3,-((x-2)/4))) + /(4*gauss_b(1/2,7/6,4/3,-((x-2)/4))*%c + +4*gauss_a(1/2,7/6,4/3,-((x-2)/4))) ) ]; [method,ode_check(eqn,ans[1])]; [riccati,0]; Index: rtestode_kamke_1_3.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_1_3.mac,v retrieving revision 1.5 retrieving revision 1.6 diff -u -d -r1.5 -r1.6 --- rtestode_kamke_1_3.mac 1 Dec 2007 21:55:14 -0000 1.5 +++ rtestode_kamke_1_3.mac 26 Dec 2007 03:42:56 -0000 1.6 @@ -18,7 +18,7 @@ /* 201 */ (pn_(201),ans:contrib_ode(eqn:2*f(x)*'diff(y,x,1)+2*f(x)*y^2-'diff(f(x),x,1)*y-2*f(x)^2,y,x)); -[y=sqrt(f(x))*(%c*sinh('integrate(sqrt(f(x)),x))+%i*cosh('integrate(sqrt(f(x)),x)))/(%i*sinh(integrate(sqrt(f(x)),x))+%c*cosh('integrate(sqrt(f(x)),x)))]; +[y=sqrt(f(x))*(%c*sinh('integrate(sqrt(f(x)),x))+%i*cosh('integrate(sqrt(f(x)),x)))/(%i*sinh('integrate(sqrt(f(x)),x))+%c*cosh('integrate(sqrt(f(x)),x)))]; [method,ode_check(eqn,ans[1])]; [riccati,0]; Index: rtestode_kamke_2_1.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_2_1.mac,v retrieving revision 1.6 retrieving revision 1.7 diff -u -d -r1.6 -r1.7 --- rtestode_kamke_2_1.mac 20 Oct 2007 13:30:17 -0000 1.6 +++ rtestode_kamke_2_1.mac 26 Dec 2007 03:42:56 -0000 1.7 @@ -106,8 +106,8 @@ /* Kamke 2. 12 */ (pn_(12),ans:contrib_ode(eqn:'diff(y,x,2)-(x^2+a)*y,y,x)); -[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)]; +[y = %k2*kummer_u(1/4-(a/4),1/2,-x^2)*%e^(x^2/2) + +%k1*kummer_m(1/4-(a/4),1/2,-x^2)*%e^(x^2/2)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -235,8 +235,8 @@ /* 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)); -[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)]; +[y = %k2*kummer_u(-(c/(4*b))-(a^2/(16*b))+1/4,1/2,-b*x^2)*%e^(b*x^2/2-a*x/2) + +%k1*kummer_m(-(c/(4*b))-(a^2/(16*b))+1/4,1/2,-b*x^2)*%e^(b*x^2/2-a*x/2)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -260,13 +260,15 @@ /* Kamke 2. 41 */ (pn_(41),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)+(n+1)*y,y,x)); -[y = %k2*kummer_u((n+1)/2,1/2,-x^2/2)+%k1*kummer_m((n+1)/2,1/2,-x^2/2)]; +[y = %k2*kummer_u(n/2+1/2,1/2,-(x^2/2)) + +%k1*kummer_m(n/2+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)); -[y = %k2*kummer_u(-n/2,1/2,-x^2/2)+%k1*kummer_m(-n/2,1/2,-x^2/2)]; +[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]; @@ -279,21 +281,21 @@ /* Kamke 2. 44 */ (pn_(44),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)-a*y,y,x)); -[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)]; +[y = %k2*kummer_u(1/2-(a/2),1/2,-(x^2/2))*%e^(x^2/2) + +%k1*kummer_m(1/2-(a/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)); -[y = %k2*%e^x+%k1*erf(%i*(x-2)/sqrt(2))*%e^(x-2)]; +[y = %k2*%e^(x-2)*erf(%i*x/sqrt(2)-2*%i/sqrt(2))+%k1*%e^x]; [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)); -[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]; +[y = %k2*kummer_u(a/4+1/2,1/2,-x^2)*%e^x^2 + +%k1*kummer_m(a/4+1/2,1/2,-x^2)*%e^x^2]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -305,8 +307,8 @@ /* 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)); -[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)]; +[y = %k2*kummer_u(n/2+1/2,1/2,-x^2)*%e^(3*x^2/2) + +%k1*kummer_m(n/2+1/2,1/2,-x^2)*%e^(3*x^2/2)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -333,7 +335,7 @@ /* 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)); -[y = %k2*%e^-(a*x^2/2+sqrt(a)*x)+%k1*%e^(x*(2*sqrt(a)-a*x)/2)]; +[y = %k2*%e^(sqrt(a)*x-(a*x^2/2))+%k1*%e^(-(a*x^2/2)-sqrt(a)*x)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -358,8 +360,7 @@ /* 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)); -[y = %k2*%e^(x*(x^2+3)/3)*'integrate(%e^(-x^3/3-2*x),x) - +%k1*%e^(x*(x^2+3)/3)]; +[y=%k2*%e^(x^3/3+x)*'integrate(%e^(-(x^3/3)-2*x),x)+%k1*%e^(x^3/3+x)]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -505,8 +506,8 @@ /* Kamke 2. 87 */ (pn_(87),ans:contrib_ode(eqn:4*'diff(y,x,2)-(x^2+a)*y,y,x)); -[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)]; +[y = %k2*kummer_u(1/4-(a/8),1/2,-(x^2/2))*%e^(x^2/4) + +%k1*kummer_m(1/4-(a/8),1/2,-(x^2/2))*%e^(x^2/4)]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -516,10 +517,10 @@ /* 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)); -[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))]; +[y=%k2*kummer_u(d/2+1/2,1/2,-(x^2/(2*a))-(c*x/a)+b*x-(c^2/(2*a))+b*c-(a*b^2/2)) + *%e^(x^2/(2*a)+c*x/a+c^2/(4*a)-(b*c/2)+a*b^2/4) + +%k1*kummer_m(d/2+1/2,1/2,-(x^2/(2*a))-(c*x/a)+b*x-(c^2/(2*a))+b*c-(a*b^2/2)) + *%e^(x^2/(2*a)+c*x/a+c^2/(4*a)-(b*c/2)+a*b^2/4)]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -571,8 +572,8 @@ /* Kamke 2. 96 */ (pn_(96),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1)+(x+a)*y,y,x)); -[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)]; +[y = %k2*kummer_u(1/2-((%i*a)/2),1,-2*%i*x)*%e^(%i*x) + +%k1*kummer_m(1/2-((%i*a)/2),1,-2*%i*x)*%e^(%i*x)]; [method, ode_check(eqn,ans[1])]; [odelin,0]; Index: rtestode_kamke_2_2.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_2_2.mac,v retrieving revision 1.11 retrieving revision 1.12 diff -u -d -r1.11 -r1.12 --- rtestode_kamke_2_2.mac 1 Dec 2007 21:55:59 -0000 1.11 +++ rtestode_kamke_2_2.mac 26 Dec 2007 03:42:56 -0000 1.12 @@ -38,8 +38,8 @@ /* Kamke 2. 105 */ (pn_(105),ans:contrib_ode(eqn:x*'diff(y,x,2)+a*'diff(y,x,1)+b*x*y,y,x)); - [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)]; + [y = %k2*bessel_y(1/2-(a/2),-sqrt(b)*x)*x^(1/2-(a/2)) + +%k1*bessel_j(1/2-(a/2),-sqrt(b)*x)*x^(1/2-(a/2))]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -85,7 +85,7 @@ /* 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));\ -[y = %k2*%e^(2*x)+%k1*(3*x+1)*%e^-x]; +[y = %k2*%e^(2*x)+%k1*(3*x*%e^-x+%e^-x)]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -108,8 +108,8 @@ /* 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)); -[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]; +[y = kummer_u(6*17^-(1/2),0,-sqrt(17)*x)*%k2*%e^(sqrt(17)*x/2+3*x/2)/x + +kummer_m(6*17^-(1/2),0,-sqrt(17)*x)*%k1*%e^(sqrt(17)*x/2+3*x/2)/x]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -120,15 +120,15 @@ /* 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)); -[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)]; +[y = %k2*kummer_u(-(b^2/(b-a))-(a*b/(b-a)),-b-a,a*x-b*x)*%e^(b*x) + +%k1*kummer_m(-(b^2/(b-a))-(a*b/(b-a)),-b-a,a*x-b*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)); -[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)]; +[y = %k2*kummer_u(m,n+m,a*x-b*x)*%e^-(a*x) + +%k1*kummer_m(m,n+m,a*x-b*x)*%e^-(a*x)]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -140,10 +140,14 @@ /* 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)); -[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)]; +[y = %k2*kummer_u(-(2*sqrt(a^2-4*c)*d/(8*c-2*a^2)) + +4*b*c/(8*c-2*a^2)+a*b*sqrt(a^2-4*c)/(8*c-2*a^2) + -(a^2*b/(8*c-2*a^2)),b,-sqrt(a^2-4*c)*x) + *%e^(sqrt(a^2-4*c)*x/2-(a*x/2)) + +%k1*kummer_m(-(2*sqrt(a^2-4*c)*d/(8*c-2*a^2)) + +4*b*c/(8*c-2*a^2)+a*b*sqrt(a^2-4*c)/(8*c-2*a^2) + -(a^2*b/(8*c-2*a^2)),b,-sqrt(a^2-4*c)*x) + *%e^(sqrt(a^2-4*c)*x/2-(a*x/2))] ; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -173,7 +177,7 @@ /* 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)); -[y = %k2*kummer_u(-n/2,(2*a+1)/2,x^2)+%k1*kummer_m(-n/2,(2*a+1)/2,x^2)]; +[y = %k2*kummer_u(-(n/2),a+1/2,x^2)+%k1*kummer_m(-(n/2),a+1/2,x^2)]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -203,7 +207,7 @@ /* 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)); -[y = %k2*(4*x^3-42*x^2+150*x-183)*%e^(3*x)+%k1*%e^x]; +[y=%k2*(4*x^3*%e^(3*x)-42*x^2*%e^(3*x)+150*x*%e^(3*x)-183*%e^(3*x))+%k1*%e^x]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -226,8 +230,8 @@ 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)); -[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]; +[y = %k2*kummer_u(a/2+1,3/2,-x)*sqrt(x)*%e^x + +%k1*kummer_m(a/2+1,3/2,-x)*sqrt(x)*%e^x]; [method, ode_check(eqn,ans[1])]; [odelin,0]; forget(4*x^2+(-8*a-4)*x+1>0); @@ -260,9 +264,9 @@ [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)); -[y = %k2*kummer_u((l+2)/4,0,-x)*%e^(x/2) - +%k1*kummer_m((l+2)/4,0,-x)*%e^(x/2)]; +(pn_(137),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+(-x-2)*y+L*y+4*y,y,x)); +[y = %k2*kummer_u(L/4+1/2,0,-x)*%e^(x/2) + +%k1*kummer_m(L/4+1/2,0,-x)*%e^(x/2)]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -274,15 +278,15 @@ /* 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)); -[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)]; +[y = %k2*kummer_u(3/4-(a/8),3/2,-(x/2))*sqrt(x)*%e^(x/4) + +%k1*kummer_m(3/4-(a/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)); -[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)]; +[y = %k2*bessel_y(b/a-1,2*sqrt(c)*sqrt(x)/sqrt(a))*x^(1/2-(b/(2*a))) + +%k1*bessel_j(b/a-1,2*sqrt(c)*sqrt(x)/sqrt(a))*x^(1/2-(b/(2*a)))]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -310,18 +314,32 @@ /* 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)); -[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))]; +[y = %k2*kummer_u(2*a0*a2*sqrt(a1^2-4*a0*a2)*b2/(8*a0*a2^3-2*a1^2*a2^2) + -(a1^2*sqrt(a1^2-4*a0*a2)*b2/(8*a0*a2^3-2*a1^2*a2^2)) + -(4*a0*a1*a2*b2/(8*a0*a2^3-2*a1^2*a2^2)) + +a1^3*b2/(8*a0*a2^3-2*a1^2*a2^2) + +a1*a2*sqrt(a1^2-4*a0*a2)*b1/(8*a0*a2^3-2*a1^2*a2^2) + +4*a0*a2^2*b1/(8*a0*a2^3-2*a1^2*a2^2) + -(a1^2*a2*b1/(8*a0*a2^3-2*a1^2*a2^2)) + -(2*a2^2*sqrt(a1^2-4*a0*a2)*b0 + /(8*a0*a2^3-2*a1^2*a2^2)),b1/a2-(a1*b2/a2^2), + -(sqrt(a1^2-4*a0*a2)*x/a2)-(sqrt(a1^2-4*a0*a2)*b2/a2^2)) + *%e^(sqrt(a1^2-4*a0*a2)*x/(2*a2)-(a1*x/(2*a2)) + +sqrt(a1^2-4*a0*a2)*b2 + /(2*a2^2)) + +%k1*kummer_m(2*a0*a2*sqrt(a1^2-4*a0*a2)*b2/(8*a0*a2^3-2*a1^2*a2^2) + -(a1^2*sqrt(a1^2-4*a0*a2)*b2/(8*a0*a2^3-2*a1^2*a2^2)) + -(4*a0*a1*a2*b2/(8*a0*a2^3-2*a1^2*a2^2)) + +a1^3*b2/(8*a0*a2^3-2*a1^2*a2^2) + +a1*a2*sqrt(a1^2-4*a0*a2)*b1/(8*a0*a2^3-2*a1^2*a2^2) + +4*a0*a2^2*b1/(8*a0*a2^3-2*a1^2*a2^2) + -(a1^2*a2*b1/(8*a0*a2^3-2*a1^2*a2^2)) + -(2*a2^2*sqrt(a1^2-4*a0*a2)*b0 + /(8*a0*a2^3-2*a1^2*a2^2)),b1/a2-(a1*b2/a2^2), + -(sqrt(a1^2-4*a0*a2)*x/a2)-(sqrt(a1^2-4*a0*a2)*b2/a2^2)) + *%e^(sqrt(a1^2-4*a0*a2)*x/(2*a2)-(a1*x/(2*a2)) + +sqrt(a1^2-4*a0*a2)*b2 + /(2*a2^2))]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -389,17 +407,19 @@ /* Kamke 2. 153 */ (pn_(153),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x^2-(v-1)*v)*y,y,x)); -[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)]; +[y = %k2*bessel_y(1/2-v,-sqrt(a)*x)*sqrt(-x) + +%k1*bessel_j(1/2-v,-sqrt(a)*x)*sqrt(-x)]; [method, ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 154 */ (pn_(154),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x^2+b*x+c)*y,y,x)); -[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)]; +[y = %k2*kummer_u(sqrt(1-4*c)/2-(sqrt(-a)*b/(2*a))+1/2,sqrt(1-4*c)+1, + -2*sqrt(-a)*x)*(-x)^(sqrt(1-4*c)/2+1/2) + *%e^(sqrt(-a)*x) + +%k1*kummer_m(sqrt(1-4*c)/2-(sqrt(-a)*b/(2*a))+1/2,sqrt(1-4*c)+1, + -2*sqrt(-a)*x)*(-x)^(sqrt(1-4*c)/2+1/2) + *%e^(sqrt(-a)*x)]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -522,15 +542,14 @@ [odelin,0]; /* 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)); -[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)]; +(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)); +[y = %k2*kummer_u(-n-(a*sqrt(-L)/(2*L)),-2*n,-2*sqrt(-L)*x) + *%e^(sqrt(-L)*x)/((-x)^n*x) + +%k1*kummer_m(-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)); [y = %k2*bessel_y(sqrt(1-4*a)/2,-%i/x)*%e^-(1/x)/sqrt(-x) @@ -540,8 +559,8 @@ /* 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)); - [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)]; + [y = %k2*bessel_y(1/2-b,-(%i*a/x))*%e^(a/x)/sqrt(-x) + +%k1*bessel_j(1/2-b,-(%i*a/x))*%e^(a/x)/sqrt(-x)]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -650,10 +669,12 @@ /* 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)); -[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)]; +[y = %k2*kummer_u(-(sqrt(-4*c+a^2-2*a+1)/2)-(a/2)+3/2, + 1-sqrt(-4*c+a^2-2*a+1),-(b/x)) + *x^(sqrt(-4*c+a^2-2*a+1)/2-(a/2)+1/2)*%e^(b/x) + +%k1*kummer_m(-(sqrt(-4*c+a^2-2*a+1)/2)-(a/2)+3/2, + 1-sqrt(-4*c+a^2-2*a+1),-(b/x)) + *x^(sqrt(-4*c+a^2-2*a+1)/2-(a/2)+1/2)*%e^(b/x)]; method; odelin; @@ -663,10 +684,10 @@ /* 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)); -[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)]; +[y = %k2*kummer_u(sqrt(1-4*b)/2+a+1/2,sqrt(1-4*b)+1,-x) + *(-x)^(sqrt(1-4*b)/2+1/2) + +%k1*kummer_m(sqrt(1-4*b)/2+a+1/2,sqrt(1-4*b)+1,-x) + *(-x)^(sqrt(1-4*b)/2+1/2)]; [method, ode_check(eqn,ans[1])]; [odelin,0]; @@ -730,7 +751,7 @@ /* 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)); -[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]; +[y = %k2*bessel_y(1/2-v,-%i*x)*sqrt(-x)*%e^-x + +%k1*bessel_j(1/2-v,-%i*x)*sqrt(-x)*%e^-x]; [method, ode_check(eqn,ans[1])]; [odelin,0]; Index: rtestode_kamke_2_3.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_2_3.mac,v retrieving revision 1.11 retrieving revision 1.12 diff -u -d -r1.11 -r1.12 --- rtestode_kamke_2_3.mac 10 Dec 2007 04:53:36 -0000 1.11 +++ rtestode_kamke_2_3.mac 26 Dec 2007 03:42:56 -0000 1.12 @@ -48,48 +48,52 @@ /* 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)); -[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)]; +[y = %k2*kummer_u(4*a1*sqrt(-4*c_1+b^2-2*b+1)/(8*a1-2*a^2) + -(a^2*sqrt(-4*c_1+b^2-2*b+1)/(8*a1-2*a^2)) + -(2*sqrt(a^2-4*a1)*b1/(8*a1-2*a^2)) + +a*sqrt(a^2-4*a1)*b/(8*a1-2*a^2)+4*a1/(8*a1-2*a^2) + -(a^2/(8*a1-2*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)/2-(b/2)+1/2) + *%e^(sqrt(a^2-4*a1)*x/2-(a*x/2)) + +%k1*kummer_m(4*a1*sqrt(-4*c_1+b^2-2*b+1)/(8*a1-2*a^2) + -(a^2*sqrt(-4*c_1+b^2-2*b+1)/(8*a1-2*a^2)) + -(2*sqrt(a^2-4*a1)*b1/(8*a1-2*a^2)) + +a*sqrt(a^2-4*a1)*b/(8*a1-2*a^2)+4*a1/(8*a1-2*a^2) + -(a^2/(8*a1-2*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)/2-(b/2)+1/2) + *%e^(sqrt(a^2-4*a1)*x/2-(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)); -[y = %k2*%e^-(x^2/2)*(sqrt(2)*sqrt(%pi)*%e^(x^2/2)*erf(x/sqrt(2))-2*x)/x - +%k1/x]; +[y=%k2*(sqrt(2)*sqrt(%pi)*erf(x/sqrt(2))/x-2*%e^-(x^2/2))+%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)); -[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]; +[y = %k2*(sqrt(2)*sqrt(%pi)*%i*%e^-(x^2/2)*erf(%i*x/sqrt(2))/x^2+2/x) + +%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)); -[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)]; +[y = %k2*kummer_u(sqrt(-4*a*(-1)^n+4*a^2+1)/4-(n/2)-(a/2)+1/4, + sqrt(-4*a*(-1)^n+4*a^2+1)/2+1,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)/4-(n/2)-(a/2)+1/4, + sqrt(-4*a*(-1)^n+4*a^2+1)/2+1,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)); -[y = %k2*x^((sqrt(-3)+1)/2)*%e^-x^2+%k1*%e^-x^2/x^(sqrt(-3)/2-1/2)]; +[y = %k2*x^(sqrt(3)*%i/2+1/2)*%e^-x^2+%k1*x^(1/2-(sqrt(3)*%i/2))*%e^-x^2]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -106,12 +110,12 @@ /* 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)); -[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)]; +[y = %k2*kummer_u(sqrt(-4*a*(-1)^n+4*a^2+1)/4-(n/2)-(a/2)+1/4, + sqrt(-4*a*(-1)^n+4*a^2+1)/2+1,x^2) + *x^(sqrt(-4*a*(-1)^n+4*a^2+1)/2+1/2)*%e^-(x^2/2) + +%k1*kummer_m(sqrt(-4*a*(-1)^n+4*a^2+1)/4-(n/2)-(a/2)+1/4, + sqrt(-4*a*(-1)^n+4*a^2+1)/2+1,x^2) + *x^(sqrt(-4*a*(-1)^n+4*a^2+1)/2+1/2)*%e^-(x^2/2)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -173,32 +177,35 @@ /* Kamke 2. 225 */ (pn_(225),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,2)-x*'diff(y,x,1)+y,y,x)); -[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)]; +[y = gauss_b(2,2,5/2,1/2-(%i*x/2))*%k2*(-%i*x-1)^(1/4)*(1-%i*x)^(1/4) + *(x^2+1)^(5/4) + +gauss_a(2,2,5/2,1/2-(%i*x/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)); -[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)]; +[y = %k2*gauss_b(1-v,v,1,1/2-(%i*x/2))*sqrt(-%i*x-1)*sqrt(1-%i*x) + /sqrt(x^2+1) + +%k1*gauss_a(1-v,v,1,1/2-(%i*x/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)); -[y = %k2*(x^2+1)*%e^(2*%i*atan(x))+%k1*(x^2+1)*%e^-(2*%i*atan(x))]; +[y = %k2*(x^2*%e^(2*%i*atan(x))+%e^(2*%i*atan(x))) + +%k1*(x^2*%e^-(2*%i*atan(x))+%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)); -[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)]; +[y = %k2*gauss_b(sqrt(1-a),-sqrt(1-a),1/2,1/2-(%i*x/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),1/2,1/2-(%i*x/2))*(-%i*x-1)^(1/4) + *(1-%i*x)^(1/4)/(x^2+1)^(3/4)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -220,7 +227,7 @@ /* Kamke 2. 231 */ (pn_(231),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)-v*(v+1)*y,y,x)); -[y = %k2*gauss_b(-v-1,v,0,-((x-1)/2))+%k1*gauss_a(-v-1,v,0,-((x-1)/2))]; +[y = %k2*gauss_b(-v-1,v,0,1/2-(x/2))+%k1*gauss_a(-v-1,v,0,1/2-(x/2))]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -235,7 +242,7 @@ /* Kamke 2. 234 */ (pn_(234),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+x*'diff(y,x,1)+2,y,x)); [y = -2*log(2*sqrt(x^2-1)+2*x)^2+2*log(2*sqrt(x^2-1)+2*x)^2/2 - +%k1*log(2*sqrt(x^2-1)+2*x)+%k2]; + +%k1*log(2*sqrt(x^2-1)+2*x)+%k2]; ode_check(eqn,ans[1]); 0; method; @@ -270,54 +277,63 @@ variationofparameters; /* 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)); - [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))]; +(pn_(239),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,2)+2*x*'diff(y,x,1)-L*y,y,x)); + [y = %k2*gauss_b(sqrt(4*L+1)/2+1/2,1/2-(sqrt(4*L+1)/2),1,1/2-(x/2)) + +%k1*gauss_a(sqrt(4*L+1)/2+1/2,1/2-(sqrt(4*L+1)/2),1,1/2-(x/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)); -[y = %k2*gauss_b(-v,v+1,1,-((x-1)/2))+%k1*gauss_a(-v,v+1,1,-((x-1)/2))]; +[y = %k2*gauss_b(-v,v+1,1,1/2-(x/2))+%k1*gauss_a(-v,v+1,1,1/2-(x/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)); -[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]; +[y = %k2*gauss_b(-v,v+1,3,1/2-(x/2))*(x-1)^2 + +%k1*gauss_a(-v,v+1,3,1/2-(x/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)); - [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]; +[y = %k2*(x^3*('integrate(8*x*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1) + -(7*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1)),x)) + /(x*%e^x+%e^x) + +3*x^2 + *('integrate(8*x*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1) + -(7*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1)),x)) + /(x*%e^x+%e^x) + +3*x + *('integrate(8*x*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1) + -(7*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1)),x)) + /(x*%e^x+%e^x) + +('integrate(8*x*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1) + -(7*%e^(2*x)/(x^4+4*x^3+6*x^2+4*x+1)),x)) + /(x*%e^x+%e^x)-2*x^2*%e^(2*x)/(x*%e^x+%e^x) + +3*x*%e^(2*x)/(x*%e^x+%e^x)) + +%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)); -[y = %k2*%e^(%i*x)/((x-1)*(x+1))+%k1*%e^-(%i*x)/((x-1)*(x+1))]; +[y = %k2/(x^2*%e^(%i*x)-%e^(%i*x))+%k1*%e^(%i*x)/(x^2-1)]; [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)); - [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]; + [y = %k2*gauss_b(-v,v+1,n+1,1/2-(x/2))/(x+1)^n + +%k1*gauss_a(-v,v+1,n+1,1/2-(x/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)); -[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]; +[y = %k2*gauss_b(-v,v+1,n+1,1/2-(x/2))*(x-1)^n + +%k1*gauss_a(-v,v+1,n+1,1/2-(x/2))*(x-1)^n]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -331,7 +347,7 @@ /* 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)); - [y = %k2*(x+1)^(1-a)*'integrate((x+1)^a/((x-1)^a*(x^2+2*x+1)),x) + [y = %k2*(x+1)^(1-a)*'integrate((x+1)^a/((x-1)^a*x^2+2*(x-1)^a*x+(x-1)^a),x) +%k1*(x+1)^(1-a)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -350,7 +366,8 @@ /* 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)); -[y = %k2*(3*x^2+a^2)/((x-a)^3*(x+a)^3)+%k1/(x+a)^3]; +[y=%k2*(3*x^2/(x^6-3*a^2*x^4+3*a^4*x^2-a^6)+a^2/(x^6-3*a^2*x^4+3*a^4*x^2-a^6)) + +%k1/(x^3+3*a*x^2+3*a^2*x+a^3)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -362,10 +379,10 @@ /* 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)); -[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)]; +[y = %k2*gauss_b(sqrt(-4*c+a^2-2*a+1)/2+a/2-(1/2), + -(sqrt(-4*c+a^2-2*a+1)/2)+a/2-(1/2),b,-x) + +%k1*gauss_a(sqrt(-4*c+a^2-2*a+1)/2+a/2-(1/2), + -(sqrt(-4*c+a^2-2*a+1)/2)+a/2-(1/2),b,-x)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -379,20 +396,34 @@ /* 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)); - [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)]; + [y = %k2*(2*x^2*%e^(5*x) + *('integrate(7*x/(x^3*%e^(5*x)-3*x^2*%e^(5*x)+3*x*%e^(5*x) + -%e^(5*x)) + -(12/(x^3*%e^(5*x)-3*x^2*%e^(5*x)+3*x*%e^(5*x) + -%e^(5*x))),x)) + /(x*%e^(3*x)-%e^(3*x)) + -((4*x*%e^(5*x) + *('integrate(7*x/(x^3*%e^(5*x)-3*x^2*%e^(5*x)+3*x*%e^(5*x) + -%e^(5*x)) + -(12/(x^3*%e^(5*x)-3*x^2*%e^(5*x)+3*x*%e^(5*x) + -%e^(5*x))),x))) + /(x*%e^(3*x)-%e^(3*x))) + +2*%e^(5*x) + *('integrate(7*x/(x^3*%e^(5*x)-3*x^2*%e^(5*x)+3*x*%e^(5*x) + -%e^(5*x)) + -(12/(x^3*%e^(5*x)-3*x^2*%e^(5*x)+3*x*%e^(5*x) + -%e^(5*x))),x)) + /(x*%e^(3*x)-%e^(3*x))-(x^2/(x*%e^(3*x)-%e^(3*x))) + -(4*x/(x*%e^(3*x)-%e^(3*x)))) + +%k1*(x*%e^(2*x)-%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)); -[y=%k1*%e^((a+1)*log(x)+(1-a)*log(x-1))*'integrate(%e^(-(a+1)*log(x)-(1-a)*log(x-1))/(x^2-x),x)+%k2*%e^((a+1)*log(x)+(1-a)*log(x-1))]; +[y = %k1*%e^((a+1)*log(x)+(1-a)*log(x-1)) + *'integrate(%e^(-(a+1)*log(x)-(1-a)*log(x-1))/(x^2-x),x) + +%k2*%e^((a+1)*log(x)+(1-a)*log(x-1))]; ode_check(eqn,ans[1]); 0; method; @@ -414,17 +445,17 @@ /* 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)); -[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)]; + [y = %k2*gauss_b(sqrt(-4*c+a^2-2*a+1)/2+a/2-(1/2), + -(sqrt(-4*c+a^2-2*a+1)/2)+a/2-(1/2),-b,x) + +%k1*gauss_a(sqrt(-4*c+a^2-2*a+1)/2+a/2-(1/2), + -(sqrt(-4*c+a^2-2*a+1)/2)+a/2-(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)); -[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)]; +[y = %k2*gauss_b(sqrt(4*l+a^2)/2+a/2,a/2-(sqrt(4*l+a^2)/2),-b,x) + +%k1*gauss_a(sqrt(4*l+a^2)/2+a/2,a/2-(sqrt(4*l+a^2)/2),-b,x)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -442,7 +473,8 @@ /* 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)); -[y = %k2*%e^-x*'integrate((x+1)*%e^(x^2/(x+1)+x/(x+1)-1/(x+1)),x) +[y = %k2*%e^-x*'integrate(x*%e^(x^2/(x+1)+x/(x+1)-(1/(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]; @@ -487,27 +519,106 @@ /* 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)); -[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)]; +[y = %k2*gauss_b(-v,1/2,1/2-v,x)*x^(-v-1/2) + +%k1*gauss_a(-v,1/2,1/2-v,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)); -[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)) +[y = %k2*(x^5*sqrt(x+1) + *('integrate(43*x*%e^x + /(x^6*sqrt(x+1)+12*x^5*sqrt(x+1) + +60*x^4*sqrt(x+1) + +160*x^3*sqrt(x+1) + +240*x^2*sqrt(x+1) + +192*x*sqrt(x+1)+64*sqrt(x+1)) + +36*%e^x + /(x^6*sqrt(x+1)+12*x^5*sqrt(x+1) + +60*x^4*sqrt(x+1) + +160*x^3*sqrt(x+1) + +240*x^2*sqrt(x+1) + +192*x*sqrt(x+1) + +64*sqrt(x+1)),x)) + /(x*sqrt(x+1)*%e^(3*x)+2*sqrt(x+1)*%e^(3*x)) + +10*x^4*sqrt(x+1) + *('integrate(43*x*%e^x + /(x^6*sqrt(x+1)+12*x^5*sqrt(x+1) + +60*x^4*sqrt(x+1) + +160*x^3*sqrt(x+1) + +240*x^2*sqrt(x+1) + +192*x*sqrt(x+1)+64*sqrt(x+1)) + +36*%e^x + /(x^6*sqrt(x+1)+12*x^5*sqrt(x+1) + +60*x^4*sqrt(x+1) + +160*x^3*sqrt(x+1) + +240*x^2*sqrt(x+1) + +192*x*sqrt(x+1) + +64*sqrt(x+1)),x)) + /(x*sqrt(x+1)*%e^(3*x)+2*sqrt(x+1)*%e^(3*x)) + +40*x^3*sqrt(x+1) + *('integrate(43*x*%e^x + /(x^6*sqrt(x+1)+12*x^5*sqrt(x+1) + +60*x^4*sqrt(x+1) + +160*x^3*sqrt(x+1) + +240*x^2*sqrt(x+1) + +192*x*sqrt(x+1)+64*sqrt(x+1)) + +36*%e^x + /(x^6*sqrt(x+1)+12*x^5*sqrt(x+1) + +60*x^4*sqrt(x+1) + +160*x^3*sqrt(x+1) + +240*x^2*sqrt(x+1) + +192*x*sqrt(x+1) + +64*sqrt(x+1)),x)) + /(x*sqrt(x+1)*%e^(3*x)+2*sqrt(x+1)*%e^(3*x)) + +80*x^2*sqrt(x+1) + *('integrate(43*x*%e^x + /(x^6*sqrt(x+1)+12*x^5*sqrt(x+1) + +60*x^4*sqrt(x+1) + +160*x^3*sqrt(x+1) + +240*x^2*sqrt(x+1) + +192*x*sqrt(x+1)+64*sqrt(x+1)) + +36*%e^x + /(x^6*sqrt(x+1)+12*x^5*sqrt(x+1) + +60*x^4*sqrt(x+1) + +160*x^3*sqrt(x+1) + +240*x^2*sqrt(x+1) + +192*x*sqrt(x+1) + +64*sqrt(x+1)),x)) + /(x*sqrt(x+1)*%e^(3*x)+2*sqrt(x+1)*%e^(3*x)) + +80*x*sqrt(x+1) + *('integrate(43*x*%e^x + /(x^6*sqrt(x+1)+12*x^5*sqrt(x+1) + +60*x^4*sqrt(x+1) + +160*x^3*sqrt(x+1) + +240*x^2*sqrt(x+1) + +192*x*sqrt(x+1)+64*sqrt(x+1)) + +36*%e^x + /(x^6*sqrt(x+1)+12*x^5*sqrt(x+1) + +60*x^4*sqrt(x+1) + +160*x^3*sqrt(x+1) + +240*x^2*sqrt(x+1) + +192*x*sqrt(x+1) + +64*sqrt(x+1)),x)) + /(x*sqrt(x+1)*%e^(3*x)+2*sqrt(x+1)*%e^(3*x)) + +32*sqrt(x+1) + *('integrate(43*x*%e^x + /(x^6*sqrt(x+1)+12*x^5*sqrt(x+1) + +60*x^4*sqrt(x+1) + +160*x^3*sqrt(x+1) + +240*x^2*sqrt(x+1) + +192*x*sqrt(x+1)+64*sqrt(x+1)) + +36*%e^x + /(x^6*sqrt(x+1)+12*x^5*sqrt(x+1) + +60*x^4*sqrt(x+1) + +160*x^3*sqrt(x+1) + +240*x^2*sqrt(x+1) + +192*x*sqrt(x+1) + +64*sqrt(x+1)),x)) + /(x*sqrt(x+1)*%e^(3*x)+2*sqrt(x+1)*%e^(3*x)) + +4*x^2*%e^x/(x*sqrt(x+1)*%e^(3*x)+2*sqrt(x+1)*%e^(3*x)) + +22*x*%e^x/(x*sqrt(x+1)*%e^(3*x)+2*sqrt(x+1)*%e^(3*x)) + +18*%e^x/(x*sqrt(x+1)*%e^(3*x)+2*sqrt(x+1)*%e^(3*x))) +%k1*(x+2)^4*%e^-(3*x)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -528,8 +639,8 @@ /* 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)); -[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)]; +[y = %k2*kummer_u(-m+k+1/2,1-2*m,-x)*(-x)^(1/2-m)*%e^(x/2) + +%k1*kummer_m(-m+k+1/2,1-2*m,-x)*(-x)^(1/2-m)*%e^(x/2)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -597,8 +708,8 @@ /* 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)); -[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)]; +[y = %k2*gauss_b(a/2,a/2+1/2,a,4*x)*sqrt(4*x-1) + +%k1*gauss_a(a/2,a/2+1/2,a,4*x)*sqrt(4*x-1)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -670,22 +781,30 @@ /* 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)); -[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))]; +[y = %k2*kummer_u(4*a0*a2*sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2) + /(8*a0*a2^2-2*a1^2*a2) + -(a1^2*sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2) + /(8*a0*a2^2-2*a1^2*a2)) + +a1*sqrt(a1^2-4*a0*a2)*b1/(8*a0*a2^2-2*a1^2*a2) + -(2*a2*sqrt(a1^2-4*a0*a2)*b0/(8*a0*a2^2-2*a1^2*a2)) + +4*a0*a2^2/(8*a0*a2^2-2*a1^2*a2) + -(a1^2*a2/(8*a0*a2^2-2*a1^2*a2)), + sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2)/a2+1, + -(sqrt(a1^2-4*a0*a2)*x/a2)) + *x^(sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2)/(2*a2)-(b1/(2*a2))+1/2) + *%e^(sqrt(a1^2-4*a0*a2)*x/(2*a2)-(a1*x/(2*a2))) + +%k1*kummer_m(4*a0*a2*sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2) + /(8*a0*a2^2-2*a1^2*a2) + -(a1^2*sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2) + /(8*a0*a2^2-2*a1^2*a2)) + +a1*sqrt(a1^2-4*a0*a2)*b1/(8*a0*a2^2-2*a1^2*a2) + -(2*a2*sqrt(a1^2-4*a0*a2)*b0/(8*a0*a2^2-2*a1^2*a2)) + +4*a0*a2^2/(8*a0*a2^2-2*a1^2*a2) + -(a1^2*a2/(8*a0*a2^2-2*a1^2*a2)), + sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2)/a2+1, + -(sqrt(a1^2-4*a0*a2)*x/a2)) + *x^(sqrt(-4*a2*c_0+b1^2-2*a2*b1+a2^2)/(2*a2)-(b1/(2*a2))+1/2) + *%e^(sqrt(a1^2-4*a0*a2)*x/(2*a2)-(a1*x/(2*a2)))]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -703,16 +822,42 @@ /* 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)); -[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)))]; +/*[y = %k2*(sqrt(-a)*gauss_b(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)+1/2, + 1/2-(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)),b/(2*a), + 1/2-(sqrt(-a)*x/2))*x*(1-sqrt(-a)*x)^(b/(4*a)) + /((-sqrt(-a)*x-1)^(b/(4*a))*(a*x^2+1)^(b/(4*a))) + +gauss_b(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)+1/2, + 1/2-(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)),b/(2*a), + 1/2-(sqrt(-a)*x/2)) + *x*(1-sqrt(-a)*x)^(b/(4*a)) + /((-sqrt(-a)*x-1)^(b/(4*a))*(a*x^2+1)^(b/(4*a)))) + +%k1*(sqrt(-a)*gauss_a(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)+1/2, + 1/2-(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)),b/(2*a), + 1/2-(sqrt(-a)*x/2))*x*(1-sqrt(-a)*x)^(b/(4*a)) + /((-sqrt(-a)*x-1)^(b/(4*a))*(a*x^2+1)^(b/(4*a))) + +gauss_a(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)+1/2, + 1/2-(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)),b/(2*a), + 1/2-(sqrt(-a)*x/2)) + *(1-sqrt(-a)*x)^(b/(4*a)) + /((-sqrt(-a)*x-1)^(b/(4*a))*(a*x^2+1)^(b/(4*a))))];*/ +[y = %k2*(sqrt(-a)*gauss_b(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)+1/2, + 1/2-(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)),b/(2*a), + 1/2-(sqrt(-a)*x/2))*x*(1-sqrt(-a)*x)^(b/(4*a)) + /((-sqrt(-a)*x-1)^(b/(4*a))*(a*x^2+1)^(b/(4*a))) + +gauss_b(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)+1/2, + 1/2-(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)),b/(2*a), + 1/2-(sqrt(-a)*x/2)) + *(1-sqrt(-a)*x)^(b/(4*a)) + /((-sqrt(-a)*x-1)^(b/(4*a))*(a*x^2+1)^(b/(4*a)))) + +%k1*(sqrt(-a)*gauss_a(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)+1/2, + 1/2-(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)),b/(2*a), + 1/2-(sqrt(-a)*x/2))*x*(1-sqrt(-a)*x)^(b/(4*a)) + /((-sqrt(-a)*x-1)^(b/(4*a))*(a*x^2+1)^(b/(4*a))) + +gauss_a(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)+1/2, + 1/2-(sqrt(-4*a*c+b^2-2*a*b+a^2)/(2*a)),b/(2*a), + 1/2-(sqrt(-a)*x/2)) + *(1-sqrt(-a)*x)^(b/(4*a)) + /((-sqrt(-a)*x-1)^(b/(4*a))*(a*x^2+1)^(b/(4*a))))]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -726,6 +871,6 @@ /* 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)); -[y = gauss_b(-1,2,1,-((a*x-1)/2))*%k2+gauss_a(-1,2,1,-((a*x-1)/2))*%k1]; +[y = gauss_b(-1,2,1,1/2-(a*x/2))*%k2+gauss_a(-1,2,1,1/2-(a*x/2))*%k1]; [method,ode_check(eqn,ans[1])]; [odelin,0]; Index: rtestode_kamke_2_4.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/tests/rtestode_kamke_2_4.mac,v retrieving revision 1.11 retrieving revision 1.12 diff -u -d -r1.11 -r1.12 --- rtestode_kamke_2_4.mac 10 Dec 2007 04:53:36 -0000 1.11 +++ rtestode_kamke_2_4.mac 26 Dec 2007 03:42:57 -0000 1.12 @@ -19,10 +19,10 @@ /* Kamke 2. 302 */ (pn_(302),ans:contrib_ode(eqn:(a*x+b)^2*'diff(y,x,2)*A2+(a*x+b)*'diff(y,x,1)*A1+(a*x+b)*y*A0,y,x)); -[y = %k2*(a*x+b)^((a*A2-A1)/(2*a*A2)) - *bessel_y((a*A2-A1)/(a*A2),2*sqrt(a*x+b)*sqrt(A0)/(a*sqrt(A2))) - +%k1*(a*x+b)^((a*A2-A1)/(2*a*A2)) - *bessel_j((a*A2-A1)/(a*A2),2*sqrt(a*x+b)*sqrt(A0)/(a*sqrt(A2)))]; +[y = %k2*(a*x+b)^(1/2-(A1/(2*a*A2))) + *bessel_y(1-(A1/(a*A2)),2*sqrt(a*x+b)*sqrt(A0)/(a*sqrt(A2))) + +%k1*(a*x+b)^(1/2-(A1/(2*a*A2))) + *bessel_j(1-(A1/(a*A2)),2*sqrt(a*x+b)*sqrt(A0)/(a*sqrt(A2)))]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -38,7 +38,7 @@ /* Kamke 2. 304 */ (pn_(304),ans:contrib_ode(eqn:x^3*'diff(y,x,2)+x*'diff(y,x,1)+(-2*x-3)*y,y,x)); -[y = %k2*(%e^(1/x)*'integrate(%e^-(1/x)/x,x)-2*x^3+x^2-x)/x +[y = %k2*(%e^(1/x)*('integrate(%e^-(1/x)/x,x))/x-2*x^2+x-1) +%k1*%e^(1/x)/x]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -139,7 +139,7 @@ /* Kamke 2. 321 */ (pn_(321),ans:contrib_ode(eqn:x^2*(x+1)*'diff(y,x,2)-x*(2*x+1)*'diff(y,x,1)+(2*x+1)*y,y,x)); -[y = %k2*x*(log(x)+x)+%k1*x]; +[y = %k2*(x*log(x)+x^2)+%k1*x]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -157,7 +157,7 @@ /* Kamke 2. 324 */ (pn_(324),ans:contrib_ode(eqn:'diff(y,x,2)=(5*x-4)*'diff(y,x,1)/((x-1)*x)-(9*x-6)*y/((x-1)*x^2),y,x)); -[y = %k2*x^2*(x*log(x)+1)+%k1*x^3]; +[y = %k2*(x^3*log(x)+x^2)+%k1*x^3]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -170,20 +170,20 @@ /* Kamke 2. 326 */ (pn_(326),ans:contrib_ode(eqn:'diff(y,x,2)=-'diff(y,x,1)/(x+1)-y/(x*(x+1)^2),y,x)); -[y = %k2*(x*log(x)-1)/(x+1)+%k1*x/(x+1)]; +[y = %k2*(x*log(x)/(x+1)-(1/(x+1)))+%k1*x/(x+1)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 327 */ (pn_(327),ans:contrib_ode(eqn:'diff(y,x,2)=2*'diff(y,x,1)/((x-2)*x)-y/((x-2)*x^2),y,x)); -[y = gauss_b((sqrt(2)-2)/2,sqrt(2)/2,sqrt(2)+1,x/2)*%k2*x^(1/sqrt(2)) - +gauss_a((sqrt(2)-2)/2,sqrt(2)/2,sqrt(2)+1,x/2)*%k1*x^(1/sqrt(2))]; +[y = gauss_b(1/sqrt(2)-1,sqrt(2)/2,sqrt(2)+1,x/2)*%k2*x^(1/sqrt(2)) + +gauss_a(1/sqrt(2)-1,sqrt(2)/2,sqrt(2)+1,x/2)*%k1*x^(1/sqrt(2))]; [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 328 */ (pn_(328),ans:contrib_ode(eqn:'diff(y,x,2)=2*y/((x-1)^2*x),y,x)); -[y = %k2*(2*x*log(x)-x^2+1)/(x-1)+%k1*x/(x-1)]; +[y = %k2*(2*x*log(x)/(x-1)-(x^2/(x-1))+1/(x-1))+%k1*x/(x-1)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -205,39 +205,40 @@ /* Kamke 2. 332 */ (pn_(332),ans:contrib_ode(eqn:'diff(y,x,2)='diff(y,x,1)/(x+1)-(3*x+1)*y/(4*x^2*(x+1)),y,x)); -[y = %k2*sqrt(x)*(log(x)+x)+%k1*sqrt(x)]; +[y = %k2*(sqrt(x)*log(x)+x^(3/2))+%k1*sqrt(x)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 333 */ (pn_(333),ans:contrib_ode(eqn:'diff(y,x,2)=v*(v+1)*y/(4*x^2)-(3*x-1)*'diff(y,x,1)/(2*(x-1)*x),y,x)); -[y = %k2*gauss_b(-v,1/2,-((2*v-1)/2),x)/x^(v/2) - +%k1*gauss_a(-v,1/2,-((2*v-1)/2),x)/x^(v/2)]; +[y = %k2*gauss_b(-v,1/2,1/2-v,x)/x^(v/2) + +%k1*gauss_a(-v,1/2,1/2-v,x)/x^(v/2)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 334 */ (pn_(334),ans:contrib_ode(eqn:'diff(y,x,2)=(1-(a+1)*x)*'diff(y,x,1)/((x-1)*x)-((a^2-b^2)*x+c^2)*y/(4*(x-1)*x^2),y,x)); -[y = %k2*gauss_b((c-b+a)/2,(c+b+a)/2,c+1,x)*x^(c/2) - +%k1*gauss_a((c-b+a)/2,(c+b+a)/2,c+1,x)*x^(c/2)]; +[y = %k2*gauss_b(c/2-(b/2)+a/2,c/2+b/2+a/2,c+1,x)*x^(c/2) + +%k1*gauss_a(c/2-(b/2)+a/2,c/2+b/2+a/2,c+1,x)*x^(c/2)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 335 */ (pn_(335),ans:contrib_ode(eqn:'diff(y,x,2)=-(3*x-1)*'diff(y,x,1)/(2*(x-1)*x)-(a*x+b)*y/(4*(x-1)^2*x),y,x)); -[y = %k2*gauss_b(-((2*sqrt(-b-a)-sqrt(1-4*a)-3)/4), - -((2*sqrt(-b-a)+sqrt(1-4*a)-3)/4),3/2,x)*sqrt(x) +[y = %k2*gauss_b(-(sqrt(-b-a)/2)+sqrt(1-4*a)/4+3/4, + -(sqrt(-b-a)/2)-(sqrt(1-4*a)/4)+3/4,3/2,x)*sqrt(x) /(x-1)^(sqrt(-b-a)/2) - +%k1*gauss_a(-((2*sqrt(-b-a)-sqrt(1-4*a)-3)/4), - -((2*sqrt(-b-a)+sqrt(1-4*a)-3)/4),3/2,x)*sqrt(x) + +%k1*gauss_a(-(sqrt(-b-a)/2)+sqrt(1-4*a)/4+3/4, + -(sqrt(-b-a)/2)-(sqrt(1-4*a)/4)+3/4,3/2,x)*sqrt(x) /(x-1)^(sqrt(-b-a)/2)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; /* Kamke 2. 336 */ (pn_(336),ans:contrib_ode(eqn:'diff(y,x,2)=(3*x-1)*y/((x-1)*(2*x-1)^2),y,x)); -[y = %k2*sqrt(2*x-1)*(2*x*log(2*x-1)-2*log(2*x-1)-2*x*log(x-1)+2*log(x-1)-1) - +%k1*(x-1)*sqrt(2*x-1)]; +[y = %k2*(2*x*sqrt(2*x-1)*log(2*x-1)-2*sqrt(2*x-1)*log(2*x-1) + -2*x*sqrt(2*x-1)*log(x-1)+2*sqrt(2*x-1)*log(x-1)-sqrt(2*x-1)) + +%k1*(x*sqrt(2*x-1)-sqrt(2*x-1))]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -255,7 +256,7 @@ /* Kamke 2. 338 */ (pn_(338),ans:contrib_ode(eqn:'diff(y,x,2)=(6*x-1)*'diff(y,x,1)/(3*(x-2)*x)+y/(3*(x-2)*x^2),y,x)); -[y = %k2*x*(18*x^2-102*x+187)+%k1*(x-2)^(17/6)*x^(1/6)]; +[y = %k2*(18*x^3-102*x^2+187*x)+%k1*(x-2)^(17/6)*x^(1/6)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -284,8 +285,8 @@ /* Kamke 2. 343 */ (pn_(343),ans:contrib_ode(eqn:'diff(y,x,2)=(b*(x+b)-(1-a)*a*x^2)*y/x^4,y,x)); -[y = %k2*kummer_u((2*a-1)/2,2*a,-2*b/x)*x*%e^(b/x)/(-x)^a - +%k1*kummer_m((2*a-1)/2,2*a,-2*b/x)*x*%e^(b/x)/(-x)^a]; +[y = %k2*kummer_u(a-(1/2),2*a,-(2*b/x))*x*%e^(b/x)/(-x)^a + +%k1*kummer_m(a-(1/2),2*a,-(2*b/x))*x*%e^(b/x)/(-x)^a]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -369,7 +370,8 @@ /* Kamke 2. 355 */ (pn_(355),ans:contrib_ode(eqn:'diff(y,x,2)=(1-x^3)*'diff(y,x,1)/(x*(x^3+1))+x*y/(x^3+1),y,x)); [y = %k2*(x+1)^(1/3)*(x^2-x+1)^(1/3) - *'integrate(x/((x+1)^(1/3)*(x^2-x+1)^(1/3)*(x^3+1)),x) + *'integrate(x/(x^3*(x+1)^(1/3)*(x^2-x+1)^(1/3) + +(x+1)^(1/3)*(x^2-x+1)^(1/3)),x) +%k1*(x+1)^(1/3)*(x^2-x+1)^(1/3)]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -386,7 +388,7 @@ /* Kamke 2. 358 */ (pn_(358),ans:contrib_ode(eqn:'diff(y,x,2)=(x^2-2)*'diff(y,x,1)/(x*(x^2-1))-(x^2-2)*y/(x^2*(x^2-1)),y,x)); -[y = %k2*x*(log(sqrt(x-1)*sqrt(x+1)+x)+log(2))+%k1*x]; +[y = %k2*(x*log(sqrt(x-1)*sqrt(x+1)+x)+log(2)*x)+%k1*x]; [method,ode_check(eqn,ans[1])]; [odelin,0]; @@ -402,7 +404,7 ... [truncated message content] |