[Maxima-commits] CVS: maxima/share/contrib/diffequations/tests rtestode_murphy_1_1.mac,NONE,1.1 rtestode_murphy_1_2.mac,NONE,1.1 rtestode_murphy_1_3.mac,NONE,1.1 rtestode_murphy_1_4.mac,NONE,1.1 rtestode_murphy_1_5.mac,NONE,1.1 rtestode_murphy_1_6.mac,NONE,1.1 rtestode_murphy_2_1.mac,NONE,1.1 rtestode_murphy_2_2.mac,NONE,1.1 rtestode_murphy_2_3.mac,NONE,1.1 rtestode_murphy_2_4.mac,NONE,1.1 rtestode_murphy_2_5.mac,NONE,1.1 Makefile,1.6,1.7 testsuite.lisp,1.3,1.4 rtestode_murphy1.mac,1.11,NONE rtestode_murphy2.mac,1.7,NONE

[David B. <bil...@us...>]

Update of /cvsroot/maxima/maxima/share/contrib/diffequations/tests
In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv32407/d/d/d

Modified Files:
	Makefile testsuite.lisp 
Added Files:
	rtestode_murphy_1_1.mac rtestode_murphy_1_2.mac 
	rtestode_murphy_1_3.mac rtestode_murphy_1_4.mac 
	rtestode_murphy_1_5.mac rtestode_murphy_1_6.mac 
	rtestode_murphy_2_1.mac rtestode_murphy_2_2.mac 
	rtestode_murphy_2_3.mac rtestode_murphy_2_4.mac 
	rtestode_murphy_2_5.mac 
Removed Files:
	rtestode_murphy1.mac rtestode_murphy2.mac 
Log Message:
Divide rtestode_murphy1.mac iand rtestode_murphy2.mac into multile files.


--- NEW FILE: rtestode_murphy_1_1.mac ---
/* ODE tests - Murphy equations 1.1 - 1.100

  Reference: 
    G M Murphy, Ordinary Differential Equations and Their 
    Solutions, Van Nostrand, 1960

  First Order and of First Degree, p 224 ff

*/
/* Don't kill(all); It messes up trigsimp */

/* Print ode number*/
(pn_(n_):=print("Murphy ODE 1.",n_),true);
true;

/*   1 */
(pn_(1),ans:contrib_ode(eqn:'diff(y,x)=a*f(x),y,x));
 [y=a*'integrate(f(x),x)+%c];
[method,ode_check(eqn,ans[1])];
[linear,0];

/*   2 */
(pn_(2),ans:contrib_ode(eqn:'diff(y,x)=x+sin(x)+y,y,x));
[y=%c*exp(x)-1-x-(cos(x)+sin(x))/2];
[method,ode_check(eqn,ans[1])];
[linear,0];

/*   3 */
(pn_(3),ans:contrib_ode(eqn:'diff(y,x)=x^2+3*cosh(x)+2*y,y,x));
[y=%e^(2*x)*(-%e^-(3*x)*(3*%e^(2*x)+1)/2-(2*x^2+2*x+1)*%e^-(2*x)/4+%c)];
ans:expand(ans[1]);
y = %c*%e^(2*x)-3*%e^x/2-%e^-x/2-x^2/2-x/2-1/4;
[method,trigrat(ode_check(eqn,ans))];
[linear,0];

/* 4 */
(pn_(4),ans:contrib_ode(eqn:'diff(y,x)=a+b*x+c*y,y,x));
[y=(-b*(c*x+1)*%e^-(c*x)/c^2-a*%e^-(c*x)/c+%c)*%e^(c*x)];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 5 */
(pn_(5),ans:contrib_ode(eqn:'diff(y,x)=a*cos(b*x+c)+k*y,y,x));
[y=%e^(k*x)*(a*%e^-(k*x)*(b*sin(b*x+c)-k*cos(b*x+c))/(k^2+b^2)+%c)];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 6 */
(pn_(6),ans:contrib_ode(eqn:'diff(y,x)=a*sin(b*x+c)+k*y,y,x));
[y=%e^(k*x)*(a*%e^-(k*x)*(-k*sin(b*x+c)-b*cos(b*x+c))/(k^2+b^2)+%c)];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 7 */
(pn_(7),assume(c>0,k>c),ans:contrib_ode(eqn:'diff(y,x)=a+b*exp(k*x)+c*y,y,x),forget(c>0,k>c),ans);
[y = %e^(c*x)*(b*%e^((k-c)*x)/(k-c)-a*%e^-(c*x)/c+%c)];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 8 */
(pn_(8),ans:contrib_ode(eqn:'diff(y,x)=x*(x^2-2*y),y,x));
[y = %e^-x^2*((x^2-1)*%e^x^2/2+%c)];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 9 NOTE: Case a=2 is special */
(pn_(9),ans:contrib_ode(eqn:'diff(y,x)=x*(exp(-x^2)-a*y),y,x));
[y = %e^-(a*x^2/2)*(%e^(a*x^2/2-x^2)/(a-2)+%c)];
[method,ode_check(eqn,ans[1])];
[linear,0];

ans:contrib_ode(eqn:'diff(y,x)=x*(exp(-x^2)-2*y),y,x);
[y=(x^2/2+%c)*%e^-x^2];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 10 */
(pn_(10),ans:contrib_ode(eqn:'diff(y,x)=x^2*(a*x^3+b*y),y,x));
[y = (%c-a*(3*b*x^3+9)*%e^-(b*x^3/3)/(3*b^2))*%e^(b*x^3/3)];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 11(i)  n#-1*/
(pn_(11),assume(not equal(n,-1)),ans:contrib_ode(eqn:'diff(y,x)=a*x^n*y,y,x),forget(not equal(n,-1)),ans);
[y = %c*%e^(a*x^(n+1)/(n+1))];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 11(ii) n=-1*/
ans:contrib_ode(eqn:'diff(y,x)=a*x^-1*y,y,x);
[y = %c*%e^(a*log(x))];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 12 */
(pn_(12),ans:contrib_ode(eqn:'diff(y,x)=cos(x)*sin(x)+y*cos(x),y,x));
[y = (%e^-sin(x)*(-sin(x)-1)+%c)*%e^sin(x)];
ans:expand(ans[1]);
y = %c*%e^sin(x)-sin(x)-1;
[method,ode_check(eqn,ans)];
[linear,0];

/* 13 */
(pn_(13),ans:contrib_ode(eqn:'diff(y,x)=exp(sin(x))+y*cos(x),y,x));
[y = (x+%c)*%e^sin(x)];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 14 */
(pn_(14),ans:contrib_ode(eqn:'diff(y,x)=y*cot(x),y,x));
[y = %c*sin(x)];
[method,trigsimp(ode_check(eqn,ans[1]))];
[linear,0];

/* 15 */
(pn_(15),ans:contrib_ode(eqn:'diff(y,x)=1-y*cot(x),y,x));
[y = (%c-cos(x))/sin(x)];
[method,trigsimp(ode_check(eqn,ans[1]))];
[linear,0];

/* 16 */
(pn_(16),ans:contrib_ode(eqn:'diff(y,x)=x*csc(x)-y*cot(x),y,x));
[y = (x^2/2+%c)/sin(x)];
[method,trigsimp(ode_check(eqn,ans[1]))];
[linear,0];

/* 17 */
(pn_(17),ans:contrib_ode(eqn:'diff(y,x)=(2*csc(2*x)+cot(x))*y,y,x));
[y = %c*sin(x)/(csc(2*x)+cot(2*x))];
[method,trigsimp(ode_check(eqn,ans[1]))];
[linear,0];

/* 18 */
(pn_(18),ans:contrib_ode(eqn:'diff(y,x)=sec(x)-y*cot(x),y,x));
[y = (%c-log(1-sin(x)^2)/2)/sin(x)];
ans:subst(cos(x)^2,1-sin(x)^2,ans[1]);
y = (%c-log(cos(x)))/sin(x);
[method,trigsimp(ode_check(eqn,ans))];
[linear,0];

/* 19 */
(pn_(19),ans:contrib_ode(eqn:'diff(y,x)=exp(x)*sin(x)+y*cot(x),y,x));
[y = (%e^x+%c)*sin(x)];
[method,trigsimp(ode_check(eqn,ans[1]))];
[linear,0];

/* 20 */
(pn_(20),ans:contrib_ode(eqn:'diff(y,x)+csc(x)+2*y*cot(x)=0,y,x));
[y = (cos(x)+%c)/sin(x)^2];
[method,trigsimp(ode_check(eqn,ans[1]))];
[linear,0];

/* 21 */
(pn_(21),ans:contrib_ode(eqn:'diff(y,x)=2*csc(2*x)*sec(x)^2-2*y*cot(2*x),y,x));
[y=(4*sin(2*x)/(sin(2*x)^2+cos(2*x)^2+2*cos(2*x)+1)+%c)/sin(2*x)];
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 22 This is wrong.  

  The error is also seen in ode2, and is due to an error evaluating
  integrate(2*COT(x)^2*COS(2*x)/(CSC(2*x)+COT(2*x)),x)
  This is bug # 1051437
*/
/* (pn_(22),ans:contrib_ode(eqn:'diff(y,x)=2*(cot(x)^2*cos(2*x)-y*csc(2*x)),y,x));
[y=(csc(2*x)+cot(2*x))*((2*log(sin(x)^2+cos(x)^2+2*cos(x)+1)+2*log(sin(x)^2+cos(x)^2-2*cos(x)+1)+cos(2*x)/2+%c)];
trigrat(ode_check(eqn,ans[1]));
cos(2*x)+1;
*/

/* 23 */
(pn_(23),ans:contrib_ode(eqn:'diff(y,x)=2*csc(2*x)*(sin(x)^3+y),y,x));
[y=(2*((sin(x)*sin(2*x)+cos(x)*cos(2*x)+cos(x))/(2*sin(2*x)^2+2*cos(2*x)^2+4*cos(2*x)+2)-(sin(2*x)*sin(3*x)+(cos(2*x)+1)*cos(3*x)+3*sin(x)*sin(2*x)+5*cos(x)*cos(2*x)+5*cos(x))/(4*sin(2*x)^2+4*cos(2*x)^2+8*cos(2*x)+4))+%c)/(csc(2*x)+cot(2*x))];
ans:map(trigrat,ans[1]);
y=-(sin(2*x)-2*%c*sin(x))/(2*cos(x));
[method,trigrat(ode_check(eqn,ans))];
[linear,0];

/* 24 */
(pn_(24),ans:contrib_ode(eqn:'diff(y,x)=2*csc(2*x)*(1-tan(x)^2+y),y,x));
[y=(-2*((3*cos(4*x)+3*cos(2*x)+2)*sin(6*x)+(-3*sin(4*x)-3*sin(2*x))*cos(6*x)+3*sin(4*x)+3*sin(2*x))/(3*sin(6*x)^2+(18*sin(4*x)+18*sin(2*x))*sin(6*x)+3*cos(6*x)^2+(18*cos(4*x)+18*cos(2*x)+6)*cos(6*x)+27*sin(4*x)^2+54*sin(2*x)*sin(4*x)+27*cos(4*x)^2+(54*cos(2*x)+18)*cos(4*x)+27*sin(2*x)^2+27*cos(2*x)^2+18*cos(2*x)+3)-2*((3*cos(2*x)+1)*sin(6*x)-3*sin(2*x)*cos(6*x)+(9*cos(2*x)+3)*sin(4*x)-9*sin(2*x)*cos(4*x))/(3*sin(6*x)^2+(18*sin(4*x)+18*sin(2*x))*sin(6*x)+3*cos(6*x)^2+(18*cos(4*x)+18*cos(2*x)+6)*cos(6*x)+27*sin(4*x)^2+54*sin(2*x)*sin(4*x)+27*cos(4*x)^2+(54*cos(2*x)+18)*cos(4*x)+27*sin(2*x)^2+27*cos(2*x)^2+18*cos(2*x)+3)-2*(cos(2*x)*sin(4*x)-sin(2*x)*cos(4*x)+sin(2*x))/(sin(4*x)^2+cos(4*x)^2-2*cos(4*x)+1)-2*sin(4*x)/(sin(4*x)^2+cos(4*x)^2-2*cos(4*x)+1)+%c)/(csc(2*x)+cot(2*x))];
ans:map(trigrat,ans[1]);
y=(%c*sin(2*x)-2)/(cos(2*x)+1);
[method,trigrat(ode_check(eqn,ans))];
[linear,0];

/* 25 */
(pn_(25),ans:contrib_ode(eqn:'diff(y,x)=y*sec(x),y,x));
[y = %c*(tan(x)+sec(x))];
[method,trigsimp(ode_check(eqn,ans[1]))];
[linear,0];

/* 26 */
(pn_(26),ans:contrib_ode(eqn:'diff(y,x)+tan(x)=(1-y)*sec(x),y,x));
[y=(x+%c)/(tan(x)+sec(x))];
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 27 */
(pn_(27),ans:contrib_ode(eqn:'diff(y,x)=y*tan(x),y,x));
[y=%c*sec(x)];
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 28 */
(pn_(28),ans:contrib_ode(eqn:'diff(y,x)=cos(x)+y*tan(x),y,x));
[y=sec(x)*((sin(2*x)/2+x)/2+%c)];
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 29 */
(pn_(29),ans:contrib_ode(eqn:'diff(y,x)=cos(x)-y*tan(x),y,x));
[y=(x+%c)/sec(x)];
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 30 */
(pn_(30),ans:contrib_ode(eqn:'diff(y,x)=sec(x)-y*tan(x),y,x));
[y=(tan(x)+%c)/sec(x)];
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 31 */
(pn_(31),ans:contrib_ode(eqn:'diff(y,x)=sin(2*x)+y*tan(x),y,x));
[y=(%c-2*cos(x)^3/3)*sec(x)];
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 32 */
(pn_(32),ans:contrib_ode(eqn:'diff(y,x)=sin(2*x)-y*tan(x),y,x));
[y=(%c-2*cos(x))/sec(x)];
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 33 */
(pn_(33),ans:contrib_ode(eqn:'diff(y,x)=sin(x)+2*y*tan(x),y,x));
[y=(%c-cos(x)^3/3)*sec(x)^2];
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 34 */
(pn_(34),ans:contrib_ode(eqn:'diff(y,x)=2*(1+sec(2*x)+y*tan(2*x)),y,x));
[y=sec(2*x)*(sin(2*x)+2*x+%c)];
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 35 */
(pn_(35),ans:contrib_ode(eqn:'diff(y,x)=csc(x)+3*y*tan(x),y,x));
[y=sec(x)^3*(log(sin(x))-sin(x)^2/2+%c)];
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 36 */
(pn_(36),ans:contrib_ode(eqn:'diff(y,x)=(a+cos(log(x))+sin(log(x)))*y,y,x));
[y=%c*%e^(x*(sin(log(x))+cos(log(x)))/2+x*(sin(log(x))-cos(log(x)))/2+a*x)];
(assume(x>0),ans:map(trigrat,ans[1]),forget(x>0),ans);
y=%c*%e^(x*sin(log(x))+a*x);
[method,trigrat(ode_check(eqn,ans))];
[linear,0];

/* 37 */
(pn_(37),ans:contrib_ode(eqn:'diff(y,x)=6*exp(2*x)-y*tanh(x),y,x));
[y=(%e^(3*x)+3*%e^x+%c)/cosh(x)];
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 38 - too general*/
/* 39 - too general */

/* 41  Riccati */

/* 42  Riccati */
(pn_(42),ans:contrib_ode(eqn:'diff(y,x)+1-x=(x+y)*y,y,x));
[[y = -'diff(%u,x,1)/%u,-'diff(%u,x,1)*x+%u*(x-1)+'diff(%u,x,2) = 0]];
method;
riccati;

/* 43  Riccati */
(pn_(43),ans:contrib_ode(eqn:'diff(y,x)=(x+y)^2,y,x));
[[x=%c-atan(sqrt(%t)),y=-x-sqrt(%t)],[x=atan(sqrt(%t))+%c,y=sqrt(%t)-x]];
solve(ans[1][1],sqrt(%t));
[sqrt(%t)=-tan(x-%c)];
ans[1]:ev(ans[1][2],%[1]);
y=tan(x-%c)-x;
solve(ans[2][1],sqrt(%t));
[sqrt(%t)=tan(x-%c)];
ans[2]:ev(ans[2][2],%[1]);
y=tan(x-%c)-x;
[method,trigsimp(ode_check(eqn,ans[1])),trigsimp(ode_check(eqn,ans[2]))];
[lagrange,0,0];

/* 44  Riccati */
(pn_(44),ans:contrib_ode(eqn:'diff(y,x)=(x-y)^2,y,x));
[[x = log(sqrt(%t)+1)/2-log(sqrt(%t)-1)/2+%c,y = x-sqrt(%t)],[x=-log(sqrt(%t)+1)/2+log(sqrt(%t)-1)/2+%c,y=x+sqrt(%t)]];
solve(ans[1][2],sqrt(%t));
[sqrt(%t) = x-y];
ans[1][1],%[1];
x=log(-y+x+1)/2-log(-y+x-1)/2+%c;
expand(2*(%-%c));
2*x-2*%c = log(-y+x+1)-log(-y+x-1);
map(exp,%);
%e^(2*x-2*%c) = (-y+x+1)/(-y+x-1);
ans[1]:first(solve(%,y));
y=((x-1)*%e^(2*x-2*%c)-x-1)/(%e^(2*x-2*%c)-1);
solve(ans[2][2],sqrt(%t));
[sqrt(%t) = y-x];
ans[2][1],%[1];
x=-log(y-x+1)/2+log(y-x-1)/2+%c;
expand(2*(%-%c));
2*x-2*%c = log(y-x-1)-log(y-x+1);
map(exp,%);
%e^(2*x-2*%c)=(y-x-1)/(y-x+1);
ans[2]:first(solve(%,y));
y=((x-1)*%e^(2*x-2*%c)-x-1)/(%e^(2*x-2*%c)-1);
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[lagrange,0,0];

/* 45 */
(pn_(45),ans:contrib_ode(eqn:'diff(y,x)=3*(1-x+y)+(x-y)^2,y,x));
[-log(-y+x-1)+log(-y+x-2)+x=%c];
map(exp,ans[1]-x);
(-y+x-2)/(-y+x-1)=%e^(%c-x);
ans:first(solve(%,y));
y=((x-1)*%e^(%c-x)-x+2)/(%e^(%c-x)-1);
[method,ode_check(eqn,ans)];
[lie,0];

/* 46 riccati */
(pn_(46),ans:contrib_ode(eqn:'diff(y,x)=2*x-(1+x^2)*y+y^2,y,x));
[y = ((x^2+1)*'integrate(%e^(x^3/3+x),x)-%e^(x^3/3+x)+%c*x^2+%c)/('integrate(%e^(x^3/3+x),x)+%c)];
[method,ode_check(eqn,ans[1])];
[riccati,0];

/* 47 Riccati */
(pn_(47),ans:contrib_ode(eqn:'diff(y,x)=x*(2+x^3)-(2*x^2-y)*y,y,x));
[(x*y-x^3+1)/(y-x^2) = %c];
ans:first(solve(ans[1],y));
y = (x^3-%c*x^2-1)/(x-%c);
[method,ode_check(eqn,ans)];
[lie,0];

/* 48 Riccati 
   Fails as can't solve -'diff(z,x,2) + 2*x^2*'diff(z,x) + (-x^4+2*x+1)*z = 0
   but Lie works
*/
(pn_(48),ans:contrib_ode(eqn:'diff(y,x)=1+x*(2-x^3)+(2*x^2-y)*y,y,x));
[(log(-y+x^2+1)-log(-y+x^2-1)+2*x)/2=%c];
(ans:map(exp,2*ans[1]-2*x),ans:solve(ans,y));
[y=((x^2-1)*%e^(2*%c-2*x)-x^2-1)/(%e^(2*%c-2*x)-1)];
[method,ode_check(eqn,ans[1])];
[lie,0];

/* 49 Riccati */
(pn_(49),ans:contrib_ode(eqn:'diff(y,x)=cos(x)-(sin(x)-y)*y,y,x));
[y = (%e^cos(x)*sin(x)*'integrate(%e^-cos(x),x)+%c*%e^cos(x)*sin(x)-1)/(%e^cos(x)*'integrate(%e^-cos(x),x)+%c*%e^cos(x))];
[method,ode_check(eqn,ans[1])];
[riccati,0];

/* 50 Riccati 
   Fails as can't solve 'diff(z,x,2) - sin(2*x)*'diff(z,x) + cos(2*x)*z = 0
*/
(pn_(50),ans:contrib_ode(eqn:'diff(y,x)=cos(2*x)+(sin(2*x)+y)*y,y,x));
[[y=-'diff(%u,x,1)/%u,-'diff(%u,x,1)*sin(2*x)+%u*cos(2*x)+'diff(%u,x,2) = 0]];

/* 51 Riccati */

/* 52 Riccati  ode1_riccati fails but ode1_lie succeeds */
(pn_(52),ans:contrib_ode(eqn:'diff(y,x)=(3+x-4*y)^2,y,x));
[-(log(-8*y+2*x+7)-log(-8*y+2*x+5)-4*x)/4=%c];
(ans:-4*ans[1]+4*x,ans:map(exp,ans),ans:first(solve(ans,y)));
 y = ((2*x+5)*%e^(4*x-4*%c)-2*x-7)/(8*%e^(4*x-4*%c)-8);
[method,ode_check(eqn,ans)];
[lie,0];

/* 53 Riccati */
(pn_(53),ans:contrib_ode(eqn:'diff(y,x)=(1+4*x+9*y)^2,y,x));
[-(atan((27*y+12*x+3)/2)-6*x)/6=%c];
ans:first(solve(-6*ans[1]+6*x,y));
y=(2*tan(6*x-6*%c)-12*x-3)/27;
[method,trigsimp(ode_check(eqn,ans))];
[lie,0];

/* 54 Riccati */

/* 55 Riccati and separable*/
assume(a>0,b>0);
[a>0,b>0];
(pn_(55.1),ans:contrib_ode(eqn:'diff(y,x)=a+b*y^2,y,x));
[atan(sqrt(b)*y/sqrt(a))/(sqrt(a)*sqrt(b)) = x+%c];
solve(ans[1],y);
[y = sqrt(a)*tan(sqrt(a)*sqrt(b)*x+%c*sqrt(a)*sqrt(b))/sqrt(b)];
[method,trigsimp(ode_check(eqn,%[1]))];
[separable,0];
forget(a>0,b>0);
[a>0,b>0];

assume(a>0,b<0);
[a>0,b<0];
(pn_(55.2),ans:contrib_ode(eqn:'diff(y,x)=a+b*y^2,y,x));
[log(-(sqrt(a)*sqrt(-b)-b*y)/(b*y+sqrt(a)*sqrt(-b)))/(2*sqrt(a)*sqrt(-b))=x+%c];
solve(%[1],y);
[y=-(sqrt(a)*sqrt(-b)*%e^(2*sqrt(a)*sqrt(-b)*x+2*%c*sqrt(a)*sqrt(-b))+sqrt(a)*sqrt(-b))/(b*%e^(2*sqrt(a)*sqrt(-b)*x+2*%c*sqrt(a)*sqrt(-b))-b)];
[method,ode_check(eqn,%[1])];
[separable,0];
forget(a>0,b<0);
[a>0,b<0];

/* 57 Riccati */

/* 58 Riccati */

/* 61 Riccati */

/* 62 Riccati */
(pn_(62),ans:contrib_ode(eqn:'diff(y,x)=1+a*(x-y)*y,y,x));
[y = (sqrt(2)*sqrt(%pi)*a*x*%e^(a*x^2/2)*erf(sqrt(a)*x/sqrt(2))+sqrt(a)*(2*%c*a*x*%e^(a*x^2/2)+2))/(sqrt(2)*sqrt(%pi)*a*%e^(a*x^2/2)*erf(sqrt(a)*x/sqrt(2))+2*%c*a^(3/2)*%e^(a*x^2/2))];
[method,ode_check(eqn,ans[1])];
[riccati,0];

/* 63 Riccati */

/* 64 */
(pn_(64),ans:contrib_ode(eqn:'diff(y,x)=x*y*(3+y),y,x));
[-(log(y+3)-log(y))/3=x^2/2+%c];
map(exp,3*ans[1]);
y/(y+3) = %e^(3*(x^2/2+%c));
ans:first(solve(%,y));
y=-3*%e^(3*x^2/2+3*%c)/(%e^(3*x^2/2+3*%c)-1);
[method,ode_check(eqn,%)];
[separable,0];

/* 65 Riccati - Can't solve 2nd order ode */
(pn_(65),ans:contrib_ode(eqn:'diff(y,x)=1-x-x^3+(1+2*x^2)*y-x*y^2,y,x));
[[y='diff(%u,x,1)/(%u*x),%u*x^2*(-x^3-x+1)-'diff(%u,x,1)*(-x*(2*x^2+1)-1)-'diff(%u,x,2)*x=0]];

/* 66 Riccati - Can't solve 2nd order ode */
(pn_(66),ans:contrib_ode(eqn:'diff(y,x)=x*(2+x^2*y-y^2),y,x));
[[y='diff(%u,x,1)/(%u*x),-'diff(%u,x,1)*(-x^4-1)+2*%u*x^3-'diff(%u,x,2)*x=0]];

/* 67 Riccati - Can't solve 2nd order ode */
(pn_(67),ans:contrib_ode(eqn:'diff(y,x)=x+(1-2*x)*y-(1-x)*y^2,y,x));
[[y=-'diff(%u,x,1)/(%u*(x-1)),%u*(x-1)^2*x+'diff(%u,x,2)*(x-1)-'diff(%u,x,1)*((1-2*x)*(x-1)+1)=0]];

/* 68 */
(pn_(68),ans:contrib_ode(eqn:'diff(y,x)=a*x*y^2,y,x));
[-1/(a*y)=x^2/2+%c];
ans:first(solve(ans[1],y));
y=-2/(a*x^2+2*%c*a);
[method,ode_check(eqn,ans)];
[separable,0];

/* 70 Riccati */

/* 72 Riccati */
(pn_(72),ans:contrib_ode(eqn:'diff(y,x)=sin(x)*(2*sec(x)^2-y^2),y,x));
[[y='diff(%u,x,1)/(%u*sin(x)),2*%u*sec(x)^2*sin(x)^3-'diff(%u,x,2)*sin(x)+'diff(%u,x,1)*cos(x)=0]];

/* 73 Riccati */
(pn_(73),ans:contrib_ode(eqn:'diff(y,x)+4*csc(x)=(3-cot(x))*y+sin(x)*y^2,y,x));
[[y=-'diff(%u,x,1)/(%u*sin(x)),-4*%u*csc(x)*sin(x)^2-'diff(%u,x,1)*((3-cot(x))*sin(x)+cos(x))+'diff(%u,x,2)*sin(x)=0]];

/* 74 */
(pn_(74),ans:contrib_ode(eqn:'diff(y,x)=y*sec(x)+(sin(x)-1)^2,y,x));
[y = (tan(x)+sec(x))*(-8*(log(sin(x)^2/(cos(x)+1)^2+1)/4-log(sin(x)/(cos(x)+1)+1)/2+sin(x)/((cos(x)+1)*(2*sin(x)^2/(cos(x)+1)^2+2)))+8*(-log(sin(x)^2/(cos(x)+1)^2+1)/8+log(sin(x)/(cos(x)+1)+1)/4-(sin(x)^3/(cos(x)+1)^3-sin(x)^2/(cos(x)+1)^2+sin(x)/(cos(x)+1))/(4*sin(x)^4/(cos(x)+1)^4+8*sin(x)^2/(cos(x)+1)^2+4))+2*(log(sin(x)/(cos(x)+1)+1)-log(sin(x)^2/(cos(x)+1)^2+1)/2)+%c)];
/* y = (tan(x)+sec(x))/(%c+sin(x)) ??? */
[method,trigrat(ode_check(eqn,ans[1]))];
[linear,0];

/* 75 */
(pn_(75),ans:contrib_ode(eqn:'diff(y,x)+tan(x)*(1-y^2)=0,y,x));
[-(log(y+1)-log(y-1))/2 = log(sec(x))+%c];
map(exp,2*ans[1]);
(y-1)/(y+1)=%e^(2*(log(sec(x))+%c));
ans:first(solve(%,y));
y=-(%e^(2*%c)*sec(x)^2+1)/(%e^(2*%c)*sec(x)^2-1);
[method,trigrat(ode_check(eqn,ans))];
[separable,0];

/* 76 Riccati */

/* 78 Abel */
(pn_(78),ans:contrib_ode(eqn:'diff(y,x)+(a*x+y)*y^2=0,y,x));
false;

/* 79 Abel */
(pn_(79),ans:contrib_ode(eqn:'diff(y,x)+(a*exp(x)+y)*y^2=0,y,x));
false;

/* 88 */
(pn_(88),ans:contrib_ode(eqn:'diff(y,x)+2*x*y*(1+a*x^2*y^2)=0,y,x));
[y=%e^-x^2/sqrt(%c-a*(2*x^2+1)*%e^-(2*x^2)/2)];
[method,ode_check(eqn,ans[1])];
[bernoulli,0];

/* 91 */
(pn_(91),ans:contrib_ode(eqn:'diff(y,x)+(tan(x)+y^2*sec(x))*y=0,y,x));
[y=1/(sec(x)*sqrt(2*sin(x)+%c))];
[method,trigrat(ode_check(eqn,ans[1]))];
[bernoulli,0];

/* 92 */
(pn_(92),ans:contrib_ode(eqn:'diff(y,x)+y^3*sec(x)*tan(x)=0,y,x));
[1/(2*y^2)=1/cos(x)+%c];
ans:first(solve(ans[1],y));
y=-sqrt(cos(x)/(%c*cos(x)+1))/sqrt(2);
[method,trigrat(ode_check(eqn,ans))];
[separable,0];

/* 93 - Generic Abel equation */

/* 94 */
(pn_(94),ans:contrib_ode(eqn:'diff(y,x)=(tan(x)+y^3*sec(x))*y,y,x));
[y=sec(x)/(%c-3*(tan(x)^3/3+tan(x)))^(1/3)];
[method,trigrat(ode_check(eqn,ans[1]))];
[bernoulli,0];

/* 95 */
/* (pn_(95),ans:contrib_ode(eqn:'diff(y,x)=a*x^(n/(1-n))+b*y^n,y,x)); */

/* 96 - Too general */
/* 97 - Too general */
/* 98 - Too general */

/* 99 */
(pn_(99),ans:contrib_ode(eqn:'diff(y,x)=sqrt(y),y,x));
[2*sqrt(y)=x+%c];
[method,ode_check(eqn,ans[1])];
[separable,0];

/* 100 */
(pn_(100),ans:contrib_ode(eqn:'diff(y,x)=a+by+sqrt(c+d*y),y,x));
[((-2*by-2*a)*log(sqrt(d*y+c)+by+a)+2*sqrt(d*y+c))/d = x+%c];
[method,ode_check(eqn,ans[1])];
[separable,0];

--- NEW FILE: rtestode_murphy_1_2.mac ---
/* ODE tests - Murphy equations 1.101 - 1.200

  Reference: 
    G M Murphy, Ordinary Differential Equations and Their 
    Solutions, Van Nostrand, 1960

  First Order and of First Degree, p 224 ff

*/
/* Don't kill(all); It messes up trigsimp */

/* Print ode number*/
(pn_(n_):=print("Murphy ODE 1.",n_),true);
true;

/* 101 CHECKME*/
/* assume((b^2+8*a)>0,x>0);
[b^2+8*a > 0,x > 0];
(pn_(101),ans:contrib_ode(eqn:'diff(y,x)=a*x+b*sqrt(y),y,x));
[x=%c*%e^-((b*sqrt(b^2+8*a)*log(-(-16*y^2+sqrt(b^2+8*a)*(sqrt(y)*(8*x*y+(-b^2-4*a)*x^3)+2*b*x^2*y+a*b*x^4)+2*b^2*x^2*y+(-b^3-8*a*b)*x^3*sqrt(y)+(a*b^2+4*a^2)*x^4)/(16*y^2+(-4*b^2-16*a)*x^2*y+4*a^2*x^4))+(b^2+8*a)*log(-(-2*y+b*x*sqrt(y)+a*x^2)/x^2))/(2*b^2+16*a))];
forget((b^2+8*a)>0,x>0);
[b^2+8*a>0,x>0];
method;
genhom; */

/* 102 */
(pn_(102),ans:contrib_ode(eqn:'diff(y,x)+x^3=x*sqrt(x^4+4*y),y,x));
[x = %c*%e^-(log((sqrt(4*y+x^4)-x^2)/x^2)/2)];
(ans[1]/%c)^(-2)*x^2;
%c^2 = sqrt(4*y+x^4)-x^2;
(assume(x^2+%c^2>0),ans:first(solve(%,y)),forget(x^2+%c^2>0),ans);
y=(2*%c^2*x^2+%c^4)/4; /* This is correct */
method;
genhom;

/* 103 */
(pn_(103),ans:contrib_ode(eqn:'diff(y,x)+2*y*(1-x*sqrt(y))=0,y,x));
[y = %e^-(2*x)/(%c-(-x-1)*%e^-x)^2];
map(ratsimp,ans[1]);
y=1/(%c^2*%e^(2*x)+(2*%c*x+2*%c)*%e^x+x^2+2*x+1);
ans:map(factor,%);
y=1/(%c*%e^x+x+1)^2; /* This is correct */
method;
bernoulli;

/* 104 - There are three cases.  Just do first */
assume(a>0,b>0);
[a>0,b>0];
(pn_(104),ans:contrib_ode(eqn:'diff(y,x)=sqrt(a+b*y^2),y,x));
[asinh(sqrt(b)*y/sqrt(a))/sqrt(b)=x+%c];
ans:first(solve(ans[1],y));
y=sqrt(a)*sinh(sqrt(b)*(x+%c))/sqrt(b);
[method,trigsimp(ode_check(eqn,%))];
[separable,0];
forget(a>0,b>0);
[a>0,b>0];

/* 105 CHECKME*/
/*assume(a>0);
[a>0];
(pn_(105),ans:contrib_ode(eqn:'diff(y,x)=y*sqrt(a+b*y),y,x));
[log(-(2*sqrt(a)*sqrt(b*y+a)-b*y-2*a)/(b*y))/sqrt(a) = x+%c];
/* a+b*y = a*tanh(%c-x*sqrt(a)/2)^2 */
[method,ode_check(eqn,ans[1])];
[separable,0];
forget(a>0);
[a>0]; */

/* 109 CHECK THIS */
(pn_(109),ans:contrib_ode(eqn:'diff(y,x)=cos(x)^2*cos(y),y,x));
[(log(sin(y)+1)-log(sin(y)-1))/2 = (sin(2*x)+2*x)/4+%c];
map(exp,2*ans[1]);
(sin(y)+1)/(sin(y)-1) = %e^(2*((sin(2*x)+2*x)/4+%c));
/* ans:first(solve(%,y));
y=asin(%e^(sin(2*x)/2+x+2*%c)/(%e^(sin(2*x)/2+x+2*%c)-1)+1/(%e^(sin(2*x)/2+x+2*%c)-1));
[method,trigrat(ode_check(eqn,ans))];
[separable,0]; */

/* 110 */
(pn_(110),ans:contrib_ode(eqn:'diff(y,x)=sec(x)^2*cot(y)*cos(y),y,x));
[1/cos(y)=tan(x)+%c];
solve(ans[1],y);
[y=acos(1/(tan(x)+%c))];
[method,trigrat(ode_check(eqn,%[1]))];
[separable,0];

/* 114 CHECKME - Different results interactively */
/* assume(4*a^2>4*b^2);
[4*a^2>4*b^2];
(pn_(114),ans:contrib_ode(eqn:'diff(y,x)=a+b*cos(y),y,x));
[-2*sqrt(a^2-b^2)*atan(sqrt(a^2-b^2)*sin(y)/((b+a)*cos(y)+b+a))/(b^2-a^2)=x+%c];
/* Take this on faith at present */
forget(4*a^2>4*b^2);
[4*a^2>4*b^2]; */

/* assume(4*a^2<4*b^2);
[4*b^2>4*a^2];
ans:contrib_ode(eqn,y,x);
[-log(-((a-b)*sin(y)^2+sqrt(b^2-a^2)*(2*cos(y)+2)*sin(y)+(-b-a)*cos(y)^2+(-2*b-2*a)*cos(y)-b-a)/((b-a)*sin(y)^2+(-b-a)*cos(y)^2+(-2*b-2*a)*cos(y)-b-a))/sqrt(b^2-a^2)=x+%c];
/* Take this on faith at present */
forget(4*a^2<4*b^2);
[4*b^2>4*a^2]; */


/* 115  - takes "forever" in ode1_lie() */
/* (pn_(115),ans:contrib_ode(eqn:'diff(y,x)+x*(sin(2*y)-x^2*cos(y)^2)=0,y,x)); */

/* 116 */
(pn_(116),ans:contrib_ode(eqn:'diff(y,x)+tan(x)*sec(x)*cos(y)^2,y,x));
[-tan(y)=1/cos(x)+%c];
map(atan,-ans[1]); /* FIXME: solve(%[1],y) gives wrong answer */
y=-atan(1/cos(x)+%c);
[method,trigrat(ode_check(eqn,%))];
[separable,0];

/* 117 */
(pn_(117),ans:contrib_ode(eqn:'diff(y,x)=cot(x)*cot(y),y,x));
[-log(1-sin(y)^2)/2 = log(sin(x))+%c];
/* map(exp,2*ans[1]);
1/(1-sin(y)^2) = %e^(2*(log(sin(x))+%c));
ratsimp(%);
-1/(sin(y)^2-1) = %e^(2*%c)*sin(x)^2;
trigsimp(%);
1/cos(y)^2 = %e^(2*%c)*sin(x)^2;
solve(%,y);
[y = %pi-acos(%e^-%c/sin(x)),y = acos(%e^-%c/sin(x))];
[method,trigrat(ode_check(eqn,%[1])),trigrat(ode_check(eqn,%[2]))];
[separable,0,0];*/

/* 118 */
(pn_(118),ans:contrib_ode(eqn:'diff(y,x)+cot(x)*cot(y)=0,y,x));
[log(1-sin(y)^2)/2 = log(sin(x))+%c];
/* map(exp,2*ans[1]);
1-sin(y)^2 = %e^(2*(log(sin(x))+%c));
ratsimp(%);
1-sin(y)^2 = %e^(2*%c)*sin(x)^2;
trigsimp(%);
cos(y)^2 = %e^(2*%c)*sin(x)^2;
solve(%,y);
[y = %pi-acos(%e^%c*sin(x)),y = acos(%e^%c*sin(x))];
[method,trigrat(ode_check(eqn,%[1])),trigrat(ode_check(eqn,%[2]))];
[separable,0,0];*/

/* 119 */
(pn_(119),ans:contrib_ode(eqn:'diff(y,x)=sin(x)*(csc(y)-cot(y)),y,x));
[2*log(sin(y)/(cos(y)+1))-log((sin(y)^2+cos(y)^2+2*cos(y)+1)/(cos(y)^2+2*cos(y)+1))=%c-cos(x)];

/* 120 */
(pn_(120),ans:contrib_ode(eqn:'diff(y,x)=tan(x)*cot(y),y,x));
[-log(1-sin(y)^2)/2=log(sec(x))+%c];
subst(cos(y)^2,1-sin(y)^2,ans[1]);
-log(cos(y))=log(sec(x))+%c;
subst(1/cos(x),sec(x),%);
-log(cos(y))=%c-log(cos(x));
map(exp,-%);
cos(y)=%e^-%c*cos(x);
subst(%c,%e^-%c,%); /* Change to form of the constant */
cos(y)=%c*cos(x);   /* This is correct */
solve(%,y);
[y = acos(%c*cos(x))];
[method,trigsimp(ode_check(eqn,%[1]))];
[separable, 0];


/* 121 */
(pn_(121),ans:contrib_ode(eqn:'diff(y,x)+tan(x)*cot(y)=0,y,x));
[log(1-sin(y)^2)/2 = log(sec(x))+%c];
subst(cos(y)^2,1-sin(y)^2,ans[1]);
log(cos(y))=log(sec(x))+%c;
subst(1/cos(x),sec(x),%);
log(cos(y))=%c-log(cos(x));
map(exp,%);
cos(y)=%e^%c/cos(x);
subst(%c,%e^%c,%); /* Change to form of the constant */
cos(y)=%c/cos(x);  /* This is correct */
solve(%,y);
[y = acos(%c/cos(x))];
[method,trigsimp(ode_check(eqn,%[1]))];
[separable,0];

/* 122 */
(pn_(122),ans:contrib_ode(eqn:'diff(y,x)+sin(2*x)*csc(2*y)=0,y,x));
[cos(2*y)/2=%c-cos(2*x)/2];
(ans:expand(2*ans[1]),ans:subst(%c,2*%c,ans));
cos(2*y)=%c-cos(2*x); /* This is correct */
solve(%,y);
[y = -(acos(cos(2*x)-%c)-%pi)/2];
[method,trigsimp(ode_check(eqn,%[1]))];
[separable,0];

/* 123 */
(pn_(123),ans:contrib_ode(eqn:'diff(y,x)=tan(x)*(tan(y)+sec(x)*sec(y)),y,x));
false;

/* 124 */
(pn_(124),ans:contrib_ode(eqn:'diff(y,x)=cos(x)*sec(y)^2,y,x));
[(sin(2*y)+2*y)/4 = sin(x)+%c];
(ans:4*ans[1],ans:subst(%c,4*%c,expand(ans)));
sin(2*y)+2*y=4*sin(x)+%c; /* This is correct */
method;
separable;

/* 125 */
(pn_(125),ans:contrib_ode(eqn:'diff(y,x)=sec(x)^2*sec(y)^3,y,x));
[-(sin(y)^3-3*sin(y))/3=tan(x)+%c];
(ans:3*ans[1],ans:subst(%c,3*%c,expand(ans)));
3*sin(y)-sin(y)^3=3*tan(x)+%c;
trigsimp(lhs(ans))=rhs(ans);
(cos(y)^2+2)*sin(y)=3*tan(x)+%c; /* This is correct */
method;
separable;

/* 126 */
assume(a>b,4*a^2>4*b^2);
[a>b,4*a^2>4*b^2];
(pn_(126),ans:contrib_ode(eqn:'diff(y,x)=a+b*sin(y),y,x));
[2*atan((a*sin(y)+b*cos(y)+b)/(sqrt(a^2-b^2)*(cos(y)+1)))/sqrt(a^2-b^2)=x+%c];
/* Need to check this */
forget(a>b,4*a^2>4*b^2);
[a>b,4*a^2>4*b^2];

/* 127 - FIXME: ode1_lie takes "forever" */
/* (pn_(127),ans:contrib_ode(eqn:'diff(y,x)=a+b*sin(%A*x+%B*y),y,x)); */

/* 128 - FIXME: ode1_lie takes "forever" */
/* (pn_(128),ans:contrib_ode(eqn:'diff(y,x)=(1+cos(x)*sin(y))*tan(y),y,x)); */

/* 129 */
(pn_(129),ans:contrib_ode(eqn:'diff(y,x)+csc(2*x)*sin(2*y)=0,y,x));
 [(log(sin(2*y)^2+cos(2*y)^2+2*cos(2*y)+1)-log(sin(y)^2+cos(y)^2+2*cos(y)+1)-log(sin(y)^2+cos(y)^2-2*cos(y)+1))/4=%c-log(csc(2*x)+cot(2*x))/2];
map(exp,4*ans[1]);
(sin(2*y)^2+cos(2*y)^2+2*cos(2*y)+1)/((sin(y)^2+cos(y)^2-2*cos(y)+1)*(sin(y)^2+cos(y)^2+2*cos(y)+1))=%e^(4*(%c-log(csc(2*x)+cot(2*x))/2));
trigsimp(%);
(cos(2*y)+1)/(2*sin(y)^2) = %e^(4*%c)*sin(2*x)^2/(cos(2*x)^2+2*cos(2*x)+1);
subst(%c,exp(4*%c),%);
(cos(2*y)+1)/(2*sin(y)^2) = %c*sin(2*x)^2/(cos(2*x)^2+2*cos(2*x)+1);
/* Is this equivalent to tan(x)*tan(y)=C */

/* 132 CHECKME - Why the %I in the solution? */
/* Interactive solution is equivalent but different */
(pn_(132),ans:contrib_ode(eqn:'diff(y,x)=x+exp(y),y,x));
[-%e^-y*(sqrt(2)*sqrt(%pi)*%i*erf(%i*x/sqrt(2))*%e^y-2*%e^(x^2/2))/2=%c];
solve(ans[1],y);
/*[y = log(2*%c*%e^(x^2/2)/(%pi*erf(sqrt(2)*%i*x/2)^2+2*%c^2)-sqrt(2)*sqrt(%pi)*%i*%e^(x^2/2)*erf(sqrt(2)*%i*x/2)/(%pi*erf(sqrt(2)*%i*x/2)^2+2*%c^2))];*/
[y = log(2*%e^(x^2/2)/(sqrt(2)*sqrt(%pi)*%i*erf(%i*x/sqrt(2))+2*%c))];
[method,ode_check(eqn,%[1])];
[lie,0];

/* 133 */
(pn_(133),ans:contrib_ode(eqn:'diff(y,x)=exp(x+y),y,x));
[-%e^-y*(%e^(y+x)+1)=%c];
solve(expand(ans[1]),y);
[y = log(-1/(%e^x+%c))];
[method,ode_check(eqn,%[1])];
[exact,0];

/* 134 - Solution is different to Murphy */
(pn_(134),ans:contrib_ode(eqn:'diff(y,x)=exp(x)*(a+b*exp(-y)),y,x));
[(log(a*%e^y+b)-a*%e^x)/a = %c];
a*ans[1]+a*exp(x);
log(a*%e^y+b) = a*%e^x+%c*a;
map(exp,%);
a*%e^y+b = %e^(a*%e^x+%c*a);
solve(%,y);
[y = log(%e^(a*%e^x+%c*a)/a-b/a)]; 
[method,ode_check(eqn,%[1])];
[exact,0];

/* 135 */
(pn_(135),ans:contrib_ode(eqn:'diff(y,x)+y*log(x)*log(y),y,x));
[-log(log(y))=x*log(x)-x+%c];
solve(ans[1],y);
[y=%e^%e^(-x*log(x)+x-%c)];
[method,ode_check(eqn,%[1])];
[separable,0];

/* 136 - Too general?  ode1_lie runs "forever" */
/* (pn_(136),ans:contrib_ode(eqn:'diff(y,x)=x^(m-1)*y^(1-n)*f(a*x^m+b*y^n),y,x)); */

/* 137 */
(pn_(137),ans:contrib_ode(eqn:'diff(y,x)=a*f(y),y,x));
[('integrate(1/f(y),y))/a = x+%c];
method;
separable;

/* 138 - Too general */

/* 139 */
(pn_(139),ans:contrib_ode(eqn:'diff(y,x)=f(x)*g(y),y,x));
['integrate(1/g(y),y) = 'integrate(f(x),x)+%c];
method;
separable;

/* 140 */
(pn_(140),ans:contrib_ode(eqn:2*'diff(y,x)=sec(x)^2+y*sec(x)*csc(x),y,x));
[y = %e^((log(sin(x))-log(sin(x)^2-1)/2)/2)*(('integrate(sec(x)^2*%e^-((log(sin(x))-log(sin(x)^2-1)/2)/2),x))/2+%c)]; /* Mmmm */
method;
linear;

/* 141 Riccati */
(pn_(141),ans:contrib_ode(eqn:2*'diff(y,x)+2*csc(x)^2=y*csc(x)*sec(x)+y^2*sec(x)^2,y,x));
[[y=-2*'diff(%u,x,1)/(%u*sec(x)^2),-'diff(%u,x,1)*(sec(x)^2*tan(x)+csc(x)*sec(x)^3/4)-%u*csc(x)^2*sec(x)^4/4+'diff(%u,x,2)*sec(x)^2/2=0]];
method;
riccati;

/* 142 - ode1_lie runs "forever"*/
/* (pn_(142),ans:contrib_ode(eqn:2*'diff(y,x)=2*sin(y)^2*tan(y)-x*sin(2*y),y,x)); */

/* 143 */

/* 144 */
(pn_(144),ans:contrib_ode(eqn:3*'diff(y,x)=x+sqrt(x^2-3*y),y,x));
/* [x=%c*%e^((log((6*x^2*y+3*x^3*sqrt(x^2-3*y)-3*x^4)/(6*x^2*y))-3*log((-6*y+x*sqrt(x^2-3*y)+x^2)/x))/6)];*/ /* Assume correct */
[x = %c*%e^-((3*log((-6*y+x*sqrt(x^2-3*y)+x^2)/x)-log((2*x*sqrt(x^2-3*y)-x^2)/(2*x*sqrt(x^2-3*y)+2*x^2)))/6)];
method;
genhom;

/* 145 */
assume(a>0);
[a>0];
(pn_(145),ans:contrib_ode(eqn:x*'diff(y,x)=sqrt(a^2-x^2),y,x));
[y=-a*log(2*a*sqrt(a^2-x^2)/abs(x)+2*a^2/abs(x))+sqrt(a^2-x^2)+%c];
[method,ode_check(eqn,ans[1])];
[linear,0];
forget(a>0);
[a>0];

/* 146 */
(pn_(146),ans:contrib_ode(eqn:x*'diff(y,x)+x+y=0,y,x));
[y=(%c-x^2/2)/x];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 147 */
(pn_(147),ans:contrib_ode(eqn:x*'diff(y,x)+x^2-y=0,y,x));
[y=(%c-x)*x];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 148 */
(pn_(148),ans:contrib_ode(eqn:x*'diff(y,x)=x^3-y,y,x));
[y=(x^4/4+%c)/x];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 149 */
(pn_(149),ans:contrib_ode(eqn:x*'diff(y,x)=1+x^3+y,y,x));
[y=x*(x^2/2-1/x+%c)];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 150 */
assume(not(equal(m,1)));
[not equal(m,1)];
(pn_(150),ans:contrib_ode(eqn:x*'diff(y,x)=x^m+y,y,x));
[y=x*(x^(m-1)/(m-1)+%c)];
[method,ode_check(eqn,ans[1])];
[linear,0];
forget(not(equal(m,1)));
[not equal(m,1)];

/* 151 */
(pn_(151),ans:contrib_ode(eqn:x*'diff(y,x)=x*sin(x)-y,y,x));
[y=(sin(x)-x*cos(x)+%c)/x];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 152 */
(pn_(152),ans:contrib_ode(eqn:x*'diff(y,x)=x^2*sin(x)+y,y,x));
[y=x*(%c-cos(x))];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 153 */
assume(not(equal(n,-1)));
'[not equal(n,-1)];
(pn_(153),ans:contrib_ode(eqn:x*'diff(y,x)=x^n*log(x)-y,y,x));
[y=(x^(n+1)*log(x)/(n+1)-x^(n+1)/(n+1)^2+%c)/x];
[method,ode_check(eqn,ans[1])];
[linear,0];
forget(not(equal(n,-1)));
'[not equal(n,-1)];

/* 154 */
(pn_(154),ans:contrib_ode(eqn:x*'diff(y,x)=sin(x)-2*y,y,x));
[y=(sin(x)-x*cos(x)+%c)/x^2];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 155 */
(pn_(155),ans:contrib_ode(eqn:x*'diff(y,x)=a*y,y,x));
[y=%c*%e^(a*log(x))];
ans:map(radcan,ans);
[y=%c*x^a];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 156 */
assume(not(equal(a,0)),not(equal(a,1)));
'[not equal(a,0),not equal(a,1)];
(pn_(156),ans:contrib_ode(eqn:x*'diff(y,x)=1+x+a*y,y,x));
[y=(-%e^-(a*log(x))/a+%e^((1-a)*log(x))/(1-a)+%c)*%e^(a*log(x))];
ans:map(radcan,ans);
[y=((%c*a^2-%c*a)*x^a-a*x-a+1)/(a^2-a)];
[method,ode_check(eqn,ans[1])];
[linear,0];
forget(not(equal(a,0)),not(equal(a,1)));
'[not equal(a,0),not equal(a,1)];

/* 157 */
assume(not(equal(b,1)));
'[not equal(b,1)];
(pn_(157),ans:contrib_ode(eqn:x*'diff(y,x)=a*x+b*y,y,x));
[y=(a*x*%e^-(b*log(x))/(1-b)+%c)*%e^(b*log(x))];
ans:map(radcan,ans);
[y=((%c*b-%c)*x^b-a*x)/(b-1)];
[method,ode_check(eqn,ans[1])];
[linear,0];
forget(not(equal(b,1)));
'[not equal(b,1)];

/* 158 */
assume(not(equal(b,2)));
[not equal(b,2)];
(pn_(158),ans:contrib_ode(eqn:x*'diff(y,x)=a*x^2+b*y,y,x));
[y=(a*x^2*%e^-(b*log(x))/(2-b)+%c)*%e^(b*log(x))];
ans:map(radcan,ans);
[y=((%c*b-2*%c)*x^b-a*x^2)/(b-2)];
[method,ode_check(eqn,ans[1])];
[linear,0];
forget(not(equal(b,2)));
[not equal(b,2)];

/* 159 */
assume(not(equal(n,c)));
[not equal(n,c)];
(pn_(159),ans:contrib_ode(eqn:x*'diff(y,x)=a+b*x^n+c*y,y,x));
[y=%e^(c*log(x))*(-b*%e^(n*log(x)-c*log(x))/(c-n)-a*%e^-(c*log(x))/c+%c)];
ans:map(radcan,ans);
[y=(b*c*x^n+(%c*c*n-%c*c^2)*x^c-a*n+a*c)/(c*n-c^2)];
[method,ode_check(eqn,ans[1])];
[linear,0];
forget(not(equal(n,c)));
[not equal(n,c)];

/* 160 */
(pn_(160),ans:contrib_ode(eqn:x*'diff(y,x)+2+(3-x)*y=0,y,x));
[y=(%c-2*(-x^2-2*x-2)*%e^-x)*%e^x/x^3];
ans:map(radcan,ans);
[y=(%c*%e^x+2*x^2+4*x+4)/x^3];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 161 */
(pn_(161),ans:contrib_ode(eqn:x*'diff(y,x)+x+(2+a*x)*y=0,y,x));
[y=%e^-(a*x)*(%c-(a^2*x^2-2*a*x+2)*%e^(a*x)/a^3)/x^2];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 162 */
(pn_(162),ans:contrib_ode(eqn:x*'diff(y,x)+(a+b*x)*y=0,y,x));
[y=%c*%e^(-a*log(x)-b*x)];
ans:map(radcan,ans);
[y=%c*%e^-(b*x)/x^a];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 163 */
(pn_(163),ans:contrib_ode(eqn:x*'diff(y,x)=x^3+(1-2*x^2)*y,y,x));
[y=x*%e^-x^2*(%e^x^2/2+%c)];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 164 */
(pn_(164),ans:contrib_ode(eqn:x*'diff(y,x)=a*x-(1-b*x^2)*y,y,x));
[y=(%c-a*%e^-(b*x^2/2)/b)*%e^(b*x^2/2)/x];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 165 */
(pn_(165),ans:contrib_ode(eqn:x*'diff(y,x)+(2-a*x^2)*y,y,x));
[y=%c*%e^(a*x^2/2)/x^2];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 166 */
(pn_(166),ans:contrib_ode(eqn:x*'diff(y,x)+x^2=y^2,y,x));
[y=(bessel_k(1,x)*%c-bessel_i(1,x))*x/(bessel_k(0,x)*%c+bessel_i(0,x))];
[method,ode_check(eqn,ans[1])];
[riccati,0];

/* 167 */
(pn_(167),ans:contrib_ode(eqn:x*'diff(y,x)=x^2+y*(1+y),y,x));
[y=x*tan(x+%c)];
[method,trigsimp(ode_check(eqn,ans[1]))];
[riccati,0];

/* 168 */
(pn_(168),ans:contrib_ode(eqn:x*'diff(y,x)-y+y^2=x^(2/3),y,x));
[y=x^(2/3)/(x^(1/3)/tanh(3*x^(1/3)+%c)-1/3)];
[method,trigsimp(ode_check(eqn,ans[1]))];
[riccati,0];

/* 169 */
assume(a>0,b>0);
[a>0,b>0];
(pn_(169),ans:contrib_ode(eqn:x*'diff(y,x)=a+b*y^2,y,x));
[atan(sqrt(b)*y/sqrt(a))/(sqrt(a)*sqrt(b))=log(x)+%c];
solve(ans[1],y);
[y=sqrt(a)*tan(sqrt(a)*sqrt(b)*log(x)+%c*sqrt(a)*sqrt(b))/sqrt(b)];
[method,trigsimp(ode_check(eqn,%[1]))];
[separable,0];
forget(a>0,b>0);
[a>0,b>0];

assume(a<0,b>0);
[a<0,b>0];
(pn_("169ii"),ans:contrib_ode(eqn:x*'diff(y,x)=a+b*y^2,y,x));
[log(-(sqrt(-a)*sqrt(b)-b*y)/(b*y+sqrt(-a)*sqrt(b)))/(2*sqrt(-a)*sqrt(b))=log(x)+%c];
solve(ans[1],y);
[y=-(sqrt(-a)*sqrt(b)*%e^(2*sqrt(-a)*sqrt(b)*log(x)+2*%c*sqrt(-a)*sqrt(b))+sqrt(-a)*sqrt(b))/(b*%e^(2*sqrt(-a)*sqrt(b)*log(x)+2*%c*sqrt(-a)*sqrt(b))-b)];
[method,ode_check(eqn,%[1])];
[separable,0];
forget(a<0,b>0);
[a<0,b>0];

/* 170 */
assume(a>0,b>0);
[a>0,b>0];
(pn_("170i"),ans:contrib_ode(eqn:x*'diff(y,x)=a*x^2+y+b*y^2,y,x));
[y=sqrt(a)*x*tan(sqrt(a)*sqrt(b)*x+%c)/sqrt(b)];
[method,trigsimp(ode_check(eqn,ans[1]))];
[riccati,0];
forget(a>0,b>0);
[a>0,b>0];

assume(a>0,b<0);
[a>0,b<0];
(pn_("170ii"),ans:contrib_ode(eqn:x*'diff(y,x)=a*x^2+y+b*y^2,y,x));
[y=sqrt(a)*x*tanh(sqrt(a)*sqrt(-b)*x+%c)/sqrt(-b)];
[method,trigsimp(ode_check(eqn,ans[1]))];
[riccati,0];
forget(a>0,b<0);
[a>0,b<0];

/* 171 */
assume(a>0,b>0);
[a>0,b>0];
(pn_(171),ans:contrib_ode(eqn:x*'diff(y,x)=a*x^(2*n)+(n+b*y)*y,y,x));
[y=sqrt(a)*x^n*tan(sqrt(a)*sqrt(b)*x^n/n+%c)/sqrt(b)];
[method,trigsimp(ode_check(eqn,ans[1]))];
[riccati,0];
forget(a>0,b>0);
[a>0,b>0];

/* 172 Too general */

/* 173 Too general */

/* 174 */
assume(a>0);
[a>0];
(pn_(174),ans:contrib_ode(eqn:x*'diff(y,x)+a+x*y^2=0,y,x));
[y=(bessel_y(0,2*sqrt(a)*sqrt(x))*%c+bessel_j(0,2*sqrt(a)*sqrt(x)))*sqrt(a)/((bessel_y(1,2*sqrt(a)*sqrt(x))*%c+bessel_j(1,2*sqrt(a)*sqrt(x)))*sqrt(x))];
[method,ode_check(eqn,ans[1])];
[riccati,0];
forget(a>0);
[a>0];

/* 175 */
(pn_(175),ans:contrib_ode(eqn:x*'diff(y,x)+(1-x*y)*y=0,y,x));
[y=1/(x*(%c-log(x)))];
[method,ode_check(eqn,ans[1])];
[bernoulli,0];

/* 176 */
(pn_(176),ans:contrib_ode(eqn:x*'diff(y,x)=(1-x*y)*y,y,x));
[(x^2*y-2*x)/(2*y) = %c];
solve(ans[1],y);
[y = 2*x/(x^2-2*%c)];
[method,ode_check(eqn,%[1])];
[exact,0];

/* 177 */
(pn_(177),ans:contrib_ode(eqn:x*'diff(y,x)=(1+x*y)*y,y,x));
[-(x^2*y+2*x)/(2*y)=%c];
solve(ans[1],y);
[y=-2*x/(x^2+2*%c)];
[method,ode_check(eqn,%[1])];
[exact,0];

/* 178 */
(pn_(178),ans:contrib_ode(eqn:'diff(y,x)=a*x^2+(1/x-y)*y,y,x));
[y = sqrt(a)*(%c*x*sinh(sqrt(a)*x^2/2)+%i*x*cosh(sqrt(a)*x^2/2))/(%i*sinh(sqrt(a)*x^2/2)+%c*cosh(sqrt(a)*x^2/2))];
[method,ode_check(eqn,ans[1])];
[riccati,0];

/* 179 */
(pn_(179),ans:contrib_ode(eqn:x*'diff(y,x)=x^3+(1+2*x^2)*y+x*y^2,y,x));
[y = -(%c*x^3+(2*%c+2)*x)/(%c*x^2+2)];
[method,ode_check(eqn,ans[1])];
[riccati,0];

/* 180 */
(pn_(180),ans:contrib_ode(eqn:x*'diff(y,x)=y*(1+2*x*y),y,x));
[-(x^2*y+x)/y = %c];
ans:first(solve(ans[1],y));
y = -x/(x^2+%c);
[method,ode_check(eqn,ans)];
[exact,0];

/* 181 Riccati - CHECK SOLUTION
   Fails as can't solve -a*'DIFF(z,x,2)-2*a*'DIFF(z,x,1)/x-a^2*b*z = 0 
*/
(pn_(181),assume(a>0,b>0),ans:contrib_ode(eqn:x*'diff(y,x)+b*x+(2+a*x*y)*y=0,y,x),forget(a>0,b>0),ans);
[-(sqrt(a)*sqrt(b)*(atan((a^2*x*y^2+a*y+a*b*x)/(sqrt(a)*sqrt(b)))-atan(sqrt(a)*y/sqrt(b)))-a*b*x)/(a*b)=%c];

/* 182 Riccati */
ans:contrib_ode(eqn:x*'diff(y,x)+a0+a1*x+(a2+a3*x*y)*y=0,y,x);
[[y='diff(%u,x,1)/(%u*a3),%u*a3^2*(-a0/x-a1)-'diff(%u,x,1)*a2*a3/x-'diff(%u,x,2)*a3=0]];
method;
riccati;

/* 183 Riccati */
(pn_(183),ans:contrib_ode(eqn:x*'diff(y,x)+a*x^2*y^2+2*y=b,y,x));
[[y='diff(%u,x,1)/(%u*a*x),%u*a^2*b*x-'diff(%u,x,2)*a*x-'diff(%u,x,1)*a=0]];
method;
riccati;

/* 184 Riccati */
assume(not equal(m+n,0));
[not equal(n+m,0)];
(pn_(184),ans:contrib_ode(eqn:x*'diff(y,x)+x^m+(n-m)*y/2+x^n*y^2=0,y,x));
[y = -(%c*x^(n/2+m/2)*sin(2*x^(n/2+m/2)/(n+m))-x^(n/2+m/2)*cos(2*x^(n/2+m/2)/(n+m)))/(x^n*sin(2*x^(n/2+m/2)/(n+m))+%c*x^n*cos(2*x^(n/2+m/2)/(n+m)))];
[method,ode_check(eqn,ans[1])];
[riccati,0];
forget(not equal(m+n,0));
'[not equal(n+m,0)];

/* 185 */
(pn_(185),ans:contrib_ode(eqn:x*'diff(y,x)+(a+b*x^n*y)*y,y,x));
[y = %e^-(a*log(x))/(%c-b*%e^(n*log(x)-a*log(x))/(a-n))];
ans:map(radcan,ans);
[y=(n-a)/(b*x^n+(%c*n-%c*a)*x^a)];
[method,ode_check(eqn,ans[1])];
[bernoulli,0];

/* 186 */
(pn_(186),ans:contrib_ode(eqn:x*'diff(y,x)=a*x^m-b*y-c*x^n*y^2,y,x));
[[y='diff(%u,x,1)*x^(1-n)/(%u*c),-'diff(%u,x,2)*c*x^(n-1)-'diff(%u,x,1)*(b*c*x^(n-2)-c*(n-1)*x^(n-2))+%u*a*c^2*x^(2*(n-1)+m-1)=0]];
method;
riccati;

/* 187 */
(pn_(187),ans:contrib_ode(eqn:x*'diff(y,x)=2*x-y+a*x^n*(x-y)^2,y,x));
[[y=-'diff(%u,x,1)*x^(1-n)/(%u*a),%u*a^2*x^(2*(n-1))*(a*x^(n+1)+2)-'diff(%u,x,1)*(a*x^(n-1)*(-2*a*x^n-1/x)+a*(n-1)*x^(n-2))+'diff(%u,x,2)*a*x^(n-1)=0]];
method;
riccati;

/* 188 */
(pn_(188),ans:contrib_ode(eqn:x*'diff(y,x)+(1-a*y*log(x))*y=0,y,x));
[y = 1/(x*(%c-a*(-log(x)/x-1/x)))];
[method,ode_check(eqn,ans[1])];
[bernoulli,0];

/* 189 Riccati */

/* 190 */
(pn_(190),ans:contrib_ode(eqn:x*'diff(y,x)=y*(1+y^2),y,x));
[-(log(y^2+1)-2*log(y))/2 = log(x)+%c];
ans[1],%c=log(%c)/2;
-(log(y^2+1)-2*log(y))/2 = log(x)+log(%c)/2;
map(exp,2*%);
y^2/(y^2+1) = %e^(2*(log(x)+log(%c)/2));
map(radcan,%);
y^2/(y^2+1) = %c*x^2;
ans:solve(%,y);
[y = -x*sqrt(-%c/(%c*x^2-1)),y = x*sqrt(-%c/(%c*x^2-1))];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[separable,0,0];

/* 191 */
(pn_(191),ans:contrib_ode(eqn:x*'diff(y,x)+(1-x*y^2)*y=0,y,x));
[y=1/(sqrt(2/x+%c)*x)];
[method,ode_check(eqn,ans[1])];
[bernoulli,0];

/* 192 */
(pn_(191),ans:contrib_ode(eqn:x*'diff(y,x)+(1-x*y^2)*y=0,y,x));
[y = 1/(sqrt(2/x+%c)*x)];
[method,ode_check(eqn,ans[1])];
[bernoulli,0];

/* 193 */
assume(not(equal(a,0)),not(equal(a,-1)));
'[not equal(a,0),not equal(a,-1)];
(pn_(193),ans:contrib_ode(eqn:x*'diff(y,x)=a*y+b*(1+x^2)*y^3,y,x));
[y=%e^(a*log(x))/sqrt(%c-2*(b*%e^((2*a+2)*log(x))/(2*a+2)+b*%e^(2*a*log(x))/(2*a)))];
ans:map(radcan,ans);
[y=sqrt(a)*sqrt(a+1)*x^a/sqrt(x^(2*a)*((-a-1)*b-a*b*x^2)+%c*a^2+%c*a)];
[method,ode_check(eqn,ans[1])];
[bernoulli,0];
forget(not(equal(a,0)),not(equal(a,-1)));
'[not equal(a,0),not equal(a,-1)];

/* 194 */
(pn_(194),ans:contrib_ode(eqn:x*'diff(y,x)+2*y=a*x^(2*k)*y^k,y,x));
[y=(a*(1-k)*x^2/2+%c)^(1/(1-k))/x^2]; /* CHECKME */
method;
bernoulli;

/* 195 */
(pn_(195),ans:contrib_ode(eqn:x*'diff(y,x)=4*(y-sqrt(y)),y,x));
[log(sqrt(y)-1)/2=log(x)+%c];
map(exp,2*ans[1]);
sqrt(y)-1 = %e^(2*(log(x)+%c));
solve(%,y);
[y = %e^(4*%c)*x^4+2*%e^(2*%c)*x^2+1];
[method,radcan(ode_check(eqn,%[1]))];
[separable,0];

/* 196 */
(pn_(196),ans:contrib_ode(eqn:x*'diff(y,x)+2*y=sqrt(1+y^2),y,x));
['integrate(1/(sqrt(y^2+1)-2*y),y) = log(x)+%c]; /* CHECKME */
method;
separable;

/* 197 */
(pn_(197),ans:contrib_ode(eqn:x*'diff(y,x)=y+sqrt(x^2+y^2),y,x));
[x=%c*%e^(x*asinh(y/x)/abs(x))];
assume(x>0);
[x > 0];
ratsimp(ans[1]);
x = %c*%e^asinh(y/x);
map(log,%/%c);
log(x/%c) = asinh(y/x);
solve(%,y);   
[y = x*sinh(log(x/%c))];
ode_check(eqn,%[1]);
x*cosh(log(x/%c))-sqrt(x^2*sinh(log(x/%c))^2+x^2);
[method,trigsimp(%)];
[genhom,0];
forget(x>0);
[x > 0];


/* 198 */
(pn_(198),ans:contrib_ode(eqn:x*'diff(y,x)=y+sqrt(x^2-y^2),y,x));
[x = %c*%e^(x*asin(y/x)/abs(x))];
assume(x>0);
[x > 0];
ratsimp(ans[1]);
x = %c*%e^asin(y/x);
map(log,%/%c);
log(x/%c) = asin(y/x);
solve(%,y);   
[y = x*sin(log(x/%c))]; /* CHECK ME */
method;
genhom;
forget(x>0);
[x > 0];

/* 199 */
assume(x>0,y>0);
[x>0,y>0];
(pn_(199),ans:contrib_ode(eqn:x*'diff(y,x)=y+x*sqrt(x^2+y^2),y,x));
[x-asinh(y/x)=%c];
-(ans[1]-x);
asinh(y/x)=x-%c;
map(sinh,%);
y/x=sinh(x-%c);
%*x;
y=x*sinh(x-%c);
[method,trigsimp(ode_check(eqn,%))];
[lie,0];
forget(x>0,y>0);
[x>0,y>0];

/* 200 - Another long "result" from ode1_lie */
/*(pn_(200),ans:contrib_ode(eqn:x*'diff(y,x)=y-x*(x-y)*sqrt(x^2+y^2),y,x));*/

--- NEW FILE: rtestode_murphy_1_3.mac ---
/* ODE tests - Murphy equations 1.201 - 1.300

  Reference: 
    G M Murphy, Ordinary Differential Equations and Their 
    Solutions, Van Nostrand, 1960

  First Order and of First Degree, p 224 ff

*/
/* Don't kill(all); It messes up trigsimp */

/* Print ode number*/
(pn_(n_):=print("Murphy ODE 1.",n_),true);
true;

/* 201 */
assume(b>0);
[b>0];
(pn_(201),ans:contrib_ode(eqn:x*'diff(y,x)=y+a*sqrt(y^2+b*x^2),y,x));
[x=%c*%e^(x*asinh(y/(sqrt(b)*x))/(a*abs(x)))];
assume(x>0);
[x>0];
ratsimp(ans[1]);
x = %c*%e^(asinh(y/(sqrt(b)*x))/a);
map(log,%/%c);
log(x/%c) = asinh(y/(sqrt(b)*x))/a;
map(sinh,a*%);
sinh(a*log(x/%c)) = y/(sqrt(b)*x);
solve(%,y);
[y = sqrt(b)*x*sinh(a*log(x/%c))];
[method,trigsimp(ode_check(eqn,%[1]))];
[genhom,0];
forget(b>0,x>0);
[b>0,x>0];

/* 240 riccati */

/* 249 */
ans:contrib_ode(eqn:3*x*'diff(y,x)=3*x^(2/3)+(1-3*y)*y,y,x);
[y=tanh(3*x^(1/3)+%c)*x^(1/3)];
[method,trigsimp(ode_check(eqn,ans[1]))];
[riccati,0];

/* 266 */
(pn_(266),ans:contrib_ode(eqn:x^2*'diff(y,x)=a+b*x^n+x^2*y^2,y,x));
[[y=-'diff(%u,x,1)/%u,%u*(b*x^(n-2)+a/x^2)+'diff(%u,x,2)=0]];
method;
riccati;

/* 267 Riccati */
(pn_(267),ans:contrib_ode(eqn:x^2*'diff(y,x)+2+x*y*(4+x*y)=0,y,x));
[x=%c*(x*y+2)/(x*y+1)];
method;
genhom;

/* 268 */
(pn_(268),ans:contrib_ode(eqn:x^2*'diff(y,x)+2+a*x*(1-x*y)-x^2*y^2=0,y,x));
[[y=-'diff(%u,x,1)/%u,%u*(-a/x-2/x^2)-'diff(%u,x,1)*a+'diff(%u,x,2)=0]];
method;
riccati;

/* 270 */
(pn_(270),ans:contrib_ode(eqn:x^2*'diff(y,x)=a+b*x^n+c*x^2*y^2,y,x));
[[y=-'diff(%u,x,1)/(%u*c),%u*c^2*(b*x^(n-2)+a/x^2)+'diff(%u,x,2)*c=0]];
method;
riccati;

/* 271 Riccati */
(pn_(271),assume(4*a*c-b^2-2*b-1>0),ans:contrib_ode(eqn:x^2*'diff(y,x)=a+b*x*y+c*x^2*y^2,y,x),forget(4*a*c-b^2-2*b-1>0),ans);
[x=%c*%e^(2*atan((2*c*x*y+b+1)/sqrt(4*a*c-b^2-2*b-1))/sqrt(4*a*c-b^2-2*b-1))];
method;
genhom;

/* 272 */
(pn_(272),ans:contrib_ode(eqn:x^2*'diff(y,x)=a+b*x*y+c*x^4*y^2,y,x));
[[y=-'diff(%u,x,1)/(%u*c*x^2),%u*a*c^2*x^2+'diff(%u,x,2)*c*x^2-'diff(%u,x,1)*(b*c*x+2*c*x)=0]];

/* 275 - Abel eqn of 1st kind */
(pn_(275),ans:contrib_ode(eqn:x^2*'diff(y,x)=a*x^2*y^2-a*y^3,y,x));
false;

/* 276 - Abel eqn of 1st kind */
(pn_(276),ans:contrib_ode(eqn:x^2*'diff(y,x)+a*y^2+b*x^2*y^3=0,y,x));
false;

/* 300 Riccati - Lie symmetry method works */
(pn_(300),ans:contrib_ode(eqn:(1-x^2)*'diff(y,x)=1-(2*x-y)*y,y,x));
[-((log(x+1)-log(x-1))*y-x*log(x+1)+x*log(x-1)+2)/(2*y-2*x) = %c];
ans:first(solve(ans[1],y));
y = (x*log(x+1)-x*log(x-1)+2*%c*x-2)/(log(x+1)-log(x-1)+2*%c);
[method,ode_check(eqn,ans)];
[lie,0];

--- NEW FILE: rtestode_murphy_1_4.mac ---
/* ODE tests - Murphy equations 1.301 - 1.400

  Reference: 
    G M Murphy, Ordinary Differential Equations and Their 
    Solutions, Van Nostrand, 1960

  First Order and of First Degree, p 224 ff

*/
/* Don't kill(all); It messes up trigsimp */

/* Print ode number*/
(pn_(n_):=print("Murphy ODE 1.",n_),true);
true;

/* 301 Riccati */

/* 304 - Abel eqn od 1st kind */
(pn_(304),ans:contrib_ode(eqn:(1+x^2)*'diff(y,x)=1+y^2-2*x*y*(1+y^2),y,x)); 
false;

/* 311 */
(pn_(311),ans:contrib_ode(eqn:(a^2+x^2)*'diff(y,x)=a^2+3*x*y-2*y^2,y,x));
[[y = 'diff(%u,x,1)*(x^2+a^2)/(2*%u),-2*'diff(%u,x,2)/(x^2+a^2)+2*'diff(%u,x,1)*x/(x^2+a^2)^2+4*%u*a^2/(x^2+a^2)^3=0]];
method;
riccati;

/* 324 Riccati */

/* 329 Riccati */

/* 333 Riccati 
      Solution should simplify to y=x*(%c*x+sin(x))/(%c*x-sin(x)) ?
*/
(pn_(333),ans:contrib_ode(eqn:2*x^2*'diff(y,x)=2*x*y+(1-x*cot(x))*(x^2-y^2),y,x));
[y=sqrt(4*x^3*cot(x)-4*x^2)*sqrt(-x^3*cot(x)^3+3*x^2*cot(x)^2-3*x*cot(x)+1)*(%c*sin('integrate(sqrt(-x^3*cot(x)^3+3*x^2*cot(x)^2-3*x*cot(x)+1)/sqrt(4*x^3*cot(x)-4*x^2),x))-cos('integrate(sqrt(-x^3*cot(x)^3+3*x^2*cot(x)^2-3*x*cot(x)+1)/sqrt(4*x^3*cot(x)-4*x^2),x)))/((2*x^2*cot(x)^2-4*x*cot(x)+2)*sin('integrate(sqrt(-x^3*cot(x)^3+3*x^2*cot(x)^2-3*x*cot(x)+1)/sqrt(4*x^3*cot(x)-4*x^2),x))+(2*%c*x^2*cot(x)^2-4*%c*x*cot(x)+2*%c)*cos('integrate(sqrt(-x^3*cot(x)^3+3*x^2*cot(x)^2-3*x*cot(x)+1)/sqrt(4*x^3*cot(x)-4*x^2),x)))];
[method,ode_check(eqn,ans[1])];
[riccati,0];

/* 336 Riccati */
(pn_(336),ans:contrib_ode(eqn:x*(1-2*x)*'diff(y,x)=4*x-(1+4*x)*y+y^2,y,x));
[-(x*y-2*x^2)/(y-1) = %c];
ans:first(solve(ans[1],y));
y=(2*x^2+%c)/(x+%c);
[exact,ode_check(eqn,ans)];
[exact,0];

/* 337  - check this*/
(pn_(337),ans:contrib_ode(eqn:2*x*(1-x)*'diff(y,x)+x+(1-2*x)*y=0,y,x));
[y=%e^-(log(2*x^2-2*x)/2)*(sqrt(2)*log(2*sqrt(2*x^2-2*x)/sqrt(2)+2*x-1)/4+sqrt(2*x^2-2*x)/2+%c)];
[method,radcan(ode_check(eqn,ans[1]))];
[linear,0];

/* 338 */
(pn_(338),ans:contrib_ode(eqn:2*x*(1-x)*'diff(y,x)+x+(1-2*x)*y^2=0,y,x));
[[y='diff(%u,x,1)*(2*x^2-2*x)/(%u*(2*x-1)),-'diff(%u,x,1)*((2*x-1)*(4*x-2)/(2*x^2-2*x)^2-2/(2*x^2-2*x))-'diff(%u,x,2)*(2*x-1)/(2*x^2-2*x)+%u*(2*x-1)^2/((2*x-2)*(2*x^2-2*x)^2)=0]];
method;
riccati;

/* 345 - Abel eqn of 1st kind */
(pn_(345),ans:contrib_ode(eqn:(a+b*x)^2*'diff(y,x)+c*y^2+(a+b*x)*y^3=0,y,x));
false;

/* 347 */
(pn_(347),ans:contrib_ode(eqn:x^3*'diff(y,x)=3-x^2+x^2*y,y,x));
[y=x*((x^2-1)/x^3+%c)];
[method,ode_check(eqn,ans[1])];
[linear,0];

/* 348 Riccati */
(pn_(348),ans:contrib_ode(eqn:x^3*'diff(y,x)=x^4+y^2,y,x));
[x=%c*%e^-(x^2/(y-x^2))]; 
ans:first(solve(ans[1],y));
y=(log(%c/x)+1)*x^2/log(%c/x);
[method,ode_check(eqn,ans)];
[genhom,0];

/* 352 Riccati */
(pn_(352),ans:contrib_ode(eqn:x^3*'diff(y,x)+20+x^2*y*(1-x^2*y)=0,y,x));
[x=%c*%e^-((log(x^2*y+5)-log(x^2*y-4))/9)];
ans:first(solve((ans[1]/%c)^9,y));
y=-(5*x^9+4*%c^9)/(x^11-%c^9*x^2);
[method,ode_check(eqn,ans)];
[genhom,0];

/* 353 Riccati */
(pn_(353),ans:contrib_ode(eqn:x^3*'diff(y,x)+3+(3-2*x)*x^2*y-x^6*y^2=0,y,x));
[y=-(3*%e^(4*x)-%c)/(x^3*%e^(4*x)+%c*x^3)];
[method,ode_check(eqn,ans[1])];
[riccati,0];

/* 366 Riccati */

/* 371 Riccati */

/* 373 */
assume(a>0);
[a>0];
ans:contrib_ode(eqn:x^4*'diff(y,x)+a^2+x^4*y^2=0,y,x);
[y=(x*tan((%c*x-a)/x)+a)/(x^2*tan((%c*x-a)/x))];
[method,trigsimp(ode_check(eqn,ans[1]))];
[riccati,0];
forget(a>0);
[a>0];

/* 377 */
(pn_(377),ans:contrib_ode(eqn:x*(1-x^3)*'diff(y,x)=x^2+(1-2*x*y)*y,y,x));
[[y=-'diff(%u,x,1)*(x^3-1)/(2*%u),-'diff(%u,x,1)*(-2/((x^3-1)*(x^4-x))-6*x^2/(x^3-1)^2)+2*'diff(%u,x,2)/(x^3-1)-4*%u*x/(x^3-1)^3=0]];
method;
riccati;

/* 380 Riccati */

/* 383 Abel eqn of 1st kind */
(pn_(383),ans:contrib_ode(eqn:x^7*'diff(y,x)+5*x^3*y^2+2*(1+x^2)*y^3=0,y,x));
false;

/* 385 Riccati */
(pn_(385),ans:contrib_ode(eqn:x^n*'diff(y,x)=x^(2*n-1)-y^2,y,x));
[[y='diff(%u,x,1)*x^n/%u,-'diff(%u,x,2)/x^n-'diff(%u,x,1)*n*x^(-n-1)+%u*x^(-n-1) = 0]];
method;
riccati;

/* 387 Riccati */
(pn_(387),ans:contrib_ode(eqn:x^n*'diff(y,x)+x^(2*n-2)+y^2+(1-n)*x^(n-1)*y=0,y,x));
[x=%c*%e^-atan(x^(1-n)*y)];
ans:map(log,ans[1]/%c);
log(x/%c)=-atan(x^(1-n)*y);
ans:map(tan,ans);
tan(log(x/%c))=-x^(1-n)*y;
ans:first(solve(ans,y));
y=-x^(n-1)*tan(log(x/%c));
ode_check(eqn,%);
(x^(2*n)*tan(log(x/%c))^2-x^(2*n)*sec(log(x/%c))^2+x^(2*n))/x^2;
factor(%);
x^(2*n-2)*(tan(log(x/%c))^2-sec(log(x/%c))^2+1);
trigsimp(%);
0;
method;
genhom;

/* 388 Riccati */


--- NEW FILE: rtestode_murphy_1_5.mac ---
/* ODE tests - Murphy equations 1.401 - 1.500

  Reference: 
    G M Murphy, Ordinary Differential Equations and Their 
    Solutions, Van Nostrand, 1960

  First Order and of First Degree, p 224 ff

*/
/* Don't kill(all); It messes up trigsimp */

/* Print ode number*/
(pn_(n_):=print("Murphy ODE 1.",n_),true);
true;

/* 402 - Abel eqn of 1st kind.  absolute invariant K:  1/(8*a^2*b) */
(pn_(402),a:1/3,b:-1/6,ans:contrib_ode(eqn:x^(3/2)*'diff(y,x)=a+b*x^(3/2)*y^3,y,x));
 [x = %c*%e^((sqrt(x)*(2*y*log(sqrt(x)*y+1)-2*y*log(sqrt(x)*y-2))+2*log(sqrt(x)*y+1)-2*log(sqrt(x)*y-2)-6)/(3*sqrt(x)*y+3))];
/* CHECKME:  Get different soln with ode1_abel */
kill(a,b);
done;

/* 422 - Abel eqn of 2nd kind */
(pn_(422),ans:contrib_ode(eqn:y*'diff(y,x)+x^3+y=0,y,x));
false;

--- NEW FILE: rtestode_murphy_1_6.mac ---
/* ODE tests - Murphy equations 1.501 - 1.600

  Reference: 
    G M Murphy, Ordinary Differential Equations and Their 
    Solutions, Van Nostrand, 1960

  First Order and of First Degree, p 224 ff

*/
/* Don't kill(all); It messes up trigsimp */

/* Print ode number*/
(pn_(n_):=print("Murphy ODE 1.",n_),true);
true;

/* 593 - Abel eqn of 2nd kind */
(pn_(593),ans:contrib_ode(eqn:x^7*y*'diff(y,x)=2*(1+x^2)+5*x^3*y,y,x));
false;

--- NEW FILE: rtestode_murphy_2_1.mac ---
/* ODE tests - Murphy equations 2.1 - 2.100

  Reference: 
    G M Murphy, Ordinary Differential Equations and Their 
    Solutions, Van Nostrand, 1960, pp 15-23

    First Order and Second or Higher Degree, p278 ff
*/

/* Don't kill(all); It messes up trigsimp */

/* Eliminate parameter t from singular solution */
(elim_(ans,x,y,t):=block([s],s:first(solve(ans[2],t)),s:solve(ev(ans[1],s),y),s[1]),done);
done;

/* Print ode number*/
(pn_(n_):=print("Murphy ODE 2.",n_),true);
true;

/* 1 */
assume(not(equal(n,-2)));
[not(equal(n, -2))];
(pn_(1),ans:contrib_ode(eqn:'diff(y,x)^2=a*x^n,y,x));
[-((n+2)*y+2*x*sqrt(a*x^n))/(n+2)=%c];
solve(ans[1],y); 
[y=-(2*x*sqrt(a*x^n)+%c*n+2*%c)/(n+2)];
[method,ode_check(eqn,%[1])];
[lie,0];
forget(not(equal(n,-2)));
'[not(equal(n,-2))];

/* 2 */
(pn_(2),ans:contrib_ode(eqn:'diff(y,x)^2=y,y,x));
[4*y-x^2+2*%c*x-%c^2=0];
solve(ans[1],y);
[y=(x^2-2*%c*x+%c^2)/4];
[method,ode_check(eqn,%[1])];
[lagrange,0];

/* 3 */
(pn_(3),ans:contrib_ode(eqn:'diff(y,x)^2=x-y,y,x));
[[x=%c-2*(log(%t-1)+%t),y=x-%t^2]]; /* Correct */
method;
lagrange;

/* 4 Very long "answer" from ode1_lie */
/* (pn_(4),ans:contrib_ode(eqn:'diff(y,x)^2=x^2+y,y,x)); */

/* 5 Very long "answer" from ode1_lie */
/* (pn_(5),ans:contrib_ode(eqn:'diff(y,x)^2+x^2=4*y,y,x)); */

/* 6 Very long "answer" from ode1_lie */
/* (pn_(6),ans:contrib_ode(eqn:'diff(y,x)^2+3*x^2=8*y,y,x)); */

/* 7 ode1_lie takes forever */
/* (pn_(7),ans:contrib_ode(eqn:'diff(y,x)^2+a*x^2+b*y=0,y,x)); */

/* 8 */ /* CHECKME */
(pn_(8),ans:contrib_ode(eqn:'diff(y,x)^2=1+y^2,y,x));
[[x=%c-log(2*sqrt(%t^2-1)+2*%t),y=-sqrt(%t^2-1)],[x=log(2*sqrt(%t^2-1)+2*%t)+%c,y=sqrt(%t^2-1)]];
method;
lagrange;

/* 9 */
(pn_(9),ans:contrib_ode(eqn:'diff(y,x)^2=1-y^2,y,x));
[[x = asin(%t)+%c,y = -sqrt(1-%t^2)],[x=%c-asin(%t),y=sqrt(1-%t^2)]];
solve(ans[1][1],%t);
[%t = sin(x-%c)];
ans:ans[1][2],%;
y = -sqrt(1-sin(x-%c)^2); /* y = sin(x+%c) */
[method,trigsimp(ode_check(eqn,ans))];
[lagrange,0];
/* ans[2] has %c -> -%c */
/* Other solutions y=1, y=-1 */

/* 10 */
(pn_(10),ans:contrib_ode(eqn:'diff(y,x)^2=a^2-y^2,y,x));
[[x=asin(%t/abs(a))+%c,y=-sqrt(a^2-%t^2)],[x=%c-asin(%t/abs(a)),y=sqrt(a^2-%t^2)]];
solve(ans[1][1],%t);
[%t = abs(a)*sin(x-%c)];
ans:ans[1][2],%;
y=-sqrt(a^2-a^2*sin(x-%c)^2);
[method,trigsimp(ode_check(eqn,ans))];
[lagrange,0];
/* Solutions in Murphy are
     y=(a*(1-%c^2)*sin(x)+2*%c*a*cos(x))/(1+%c^2))
     y+a*sin(x)=0
     y=a, y=-a
*/

/* 11 */
(pn_(11),ans:contrib_ode(eqn:'diff(y,x)^2=a^2*y^2,y,x));
[y = %c*%e^(a*x),y = %c*%e^-(a*x)];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[factor,0,0];

/* 12 */
assume(a>0,b>0);
[a>0,b>0];
(pn_(12),ans:contrib_ode(eqn:'diff(y,x)^2=a+b*y^2,y,x));
[[x=%c-log(2*sqrt(%t^2-a)+2*%t)/sqrt(b),y=-sqrt(%t^2-a)/sqrt(b)],[x=log(2*sqrt(%t^2-a)+2*%t)/sqrt(b)+%c,y=sqrt(%t^2-a)/sqrt(b)]]; /* CHECKME */
method;
lagrange;
forget(a>0,b>0);
[a>0,b>0];

/* 13 */
(pn_(13),ans:contrib_ode(eqn:'diff(y,x)^2=x^2*y^2,y,x));
[y = %c*%e^(x^2/2),y = %c*%e^-(x^2/2)];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[factor,0,0];

/* 14 FIXME: crashes ode1_lagrange */
/* (pn_(14),ans:contrib_ode(eqn:'diff(y,x)^2=(y-1)*y^2,y,x)); */

/* 15 ode1_lagrange takes "forever" */
/* (pn_(15),ans:contrib_ode(eqn:'diff(y,x)^2=(y-a)*(y-b)*(y-c),y,x)); */

/* 16 ode1_lagrange returns a "solution" */
/* (pn_(16),ans:contrib_ode(eqn:'diff(y,x)^2=a^2*y^n,y,x)); */

/* 17 ode1_lie crashes */
/* (pn_(17),ans:contrib_ode(eqn:'diff(y,x)^2=a^2*(1-log(y)^2)*y^2,y,x)); */

/* 18 to 24 are very (too?) general */

/* 25 */
(pn_(25),ans:contrib_ode(eqn:'diff(y,x)^2+2*'diff(y,x)+x=0,y,x));
[-(3*y+sqrt(1-x)*(2*x-2)+3*x)/3 = %c];
solve(ans[1],y);
[y = -((2*sqrt(1-x)+3)*x-2*sqrt(1-x)+3*%c)/3];
[method,ode_check(eqn,%)];
[lie,0];

/* 26 */
(pn_(26),ans:contrib_ode(eqn:'diff(y,x)^2-2*'diff(y,x)+a*(x-y)=0,y,x));
[-(4*sqrt(a*y-a*x+1)+2*a*x-4)/(sqrt(2)*sqrt(3)*sqrt(-a)) = %c];
solve(ans[1],sqrt(a*y-a*x+1));
[sqrt(a*y-a*x+1) = -(2*a*x+sqrt(2)*sqrt(3)*%c*sqrt(-a)-4)/4];
%[1]^2;
a*y-a*x+1 = (2*a*x+sqrt(2)*sqrt(3)*%c*sqrt(-a)-4)^2/16;
solve(%,y);
[y = (2*a^2*x^2+2*sqrt(2)*sqrt(3)*%c*sqrt(-a)*a*x-3*%c^2*a-4*sqrt(2)*sqrt(3)*%c*sqrt(-a))/(8*a)];
[method,ode_check(eqn,%[1])];
[lie,0];
/* Murphy gives a*(C+x)^2=4*(C+y), a*y=1+a*x */

/* 27 */
(pn_(27),ans:contrib_ode(eqn:'diff(y,x)^2-2*'diff(y,x)-y^2=0,y,x));
[[x=-log(2*sqrt(%t^2-2*%t)+2*%t-2)+sqrt(%t^2-2*%t)/%t+%c,y=-sqrt(%t^2-2*%t)],[x=log(2*sqrt(%t^2-2*%t)+2*%t-2)-sqrt(%t^2-2*%t)/%t+%c,y=sqrt(%t^2-2*%t)]];
method; /* CHECKME */
lagrange;

/* 28 */
(pn_(28),ans:contrib_ode(eqn:'diff(y,x)^2-5*'diff(y,x)+6=0,y,x));
[y=3*x+%c,y=2*x+%c];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[factor,0,0];

/* 29 */
(pn_(29),ans:contrib_ode(eqn:'diff(y,x)^2-7*'diff(y,x)+12=0,y,x));
[y=4*x+%c,y=3*x+%c];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[factor,0,0];


/* 30 */
(pn_(30),ans:contrib_ode(eqn:'diff(y,x)^2+a*'diff(y,x)+b=0,y,x));
[-(2*y+sqrt(a^2-4*b)*x+a*x)/2 = %c];
solve(ans[1],y);
[y = -((sqrt(a^2-4*b)+a)*x+2*%c)/2];
[method,ode_check(eqn,%)];
[lie,0];

/* 31 */
(pn_(31),ans:contrib_ode(eqn:'diff(y,x)^2+a*'diff(y,x)+b*x=0,y,x));
[-(12*b*y+sqrt(a^2-4*b*x)*(4*b*x-a^2)+6*a*b*x)/(12*b) = %c];
solve(ans[1],y);
[y = -(sqrt(a^2-4*b*x)*(4*b*x-a^2)+6*a*b*x+12*%c*b)/(12*b)];
[method,ode_check(eqn,%)];
[lie,0];

/* 32 */
(pn_(32),ans:contrib_ode(eqn:'diff(y,x)^2+a*'diff(y,x)+b*y=0,y,x));
[[x=%c-(log(%t)*a+2*%t)/b,y = -(%t*a+%t^2)/b]]; /* Correct */
method;
lagrange;

/* 33 */
(pn_(33),ans:contrib_ode(eqn:'diff(y,x)^2+x*'diff(y,x)+1=0,y,x));
[-(4*y-4*log(2*sqrt(x^2-4)+2*x)+x*sqrt(x^2-4)+x^2)/4=%c];
solve(ans[1],y);
[y=(4*log(2*sqrt(x^2-4)+2*x)-x*sqrt(x^2-4)-x^2-4*%c)/4];
[method,ode_check(eqn,%)];
[lie,0];

/* 34 */
(pn_(34),ans:contrib_ode(eqn:'diff(y,x)^2+x*'diff(y,x)-y=0,y,x));
[y=%c*x+%c^2,y=-x^2/4];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[clairault,0,0];

/* 35 */
(pn_(35),ans:contrib_ode(eqn:'diff(y,x)^2-x*'diff(y,x)+y=0,y,x));
[y=%c*x-%c^2,y=x^2/4];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[clairault,0,0];

/* 36 */
(pn_(36),ans:contrib_ode(eqn:'diff(y,x)^2-x*'diff(y,x)-y=0,y,x));
[[x=%e^-(log(%t)/2)*(2*%e^(3*log(%t)/2)/3+%c),y=%t^2-%t*x]];
ans:radcan(%);
[[x=(2*%t^(3/2)+3*%c)/(3*sqrt(%t)),y=%t^2-%t*x]];
ans:ans,3*%c=%c; /* change constant */
[[x=(2*%t^(3/2)+%c)/(3*sqrt(%t)),y=%t^2-%t*x]];
ans[1][2]:ans[1][2],ans[1][1];
y=%t^2-sqrt(%t)*(2*%t^(3/2)+%c)/3;
expand(3*ans);
[[3*x = 2*%t+%c/sqrt(%t),3*y = %t^2-%c*sqrt(%t)]];  /* correct */
method;
lagrange;

/* 37 */
(pn_(37),ans:contrib_ode(eqn:'diff(y,x)^2+x*'diff(y,x)+x-y=0,y,x));
[[x=%e^-%t*(%c-2*(%t-1)*%e^%t),y = (%t+1)*x+%t^2]];
ans[1][2]:ans[1][2],ans[1][1];
y=(%t+1)*%e^-%t*(%c-2*(%t-1)*%e^%t)+%t^2;
expand(ans);
[[x=%c*%e^-%t-2*%t+2,y=%c*%t*%e^-%t+%c*%e^-%t-%t^2+2]]; /* correct */
method;
lagrange;

/* 38 */
(pn_(38),ans:contrib_ode(eqn:'diff(y,x)^2+(1-x)*'diff(y,x)+y=0,y,x));
[y=%c*x-%c-%c^2,y=(x^2-2*x+1)/4];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[clairault,0,0];

/* 39 */
(pn_(39),ans:contrib_ode(eqn:'diff(y,x)^2-(1+x)*'diff(y,x)+y=0,y,x));
[y=%c*x-%c^2+%c,y=(x^2+2*x+1)/4];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[clairault,0,0];

/* 40 */
(pn_(40),ans:contrib_ode(eqn:'diff(y,x)^2-(2-x)*'diff(y,x)+1-y=0,y,x));
[y=%c*x+%c^2-2*%c+1,y=-(x^2-4*x)/4];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[clairault,0,0];

/* 41 */
(pn_(41),ans:contrib_ode(eqn:'diff(y,x)^2+(a+x)*'diff(y,x)-y=0,y,x));
[y=%c*x+%c*a+%c^2,y=-(x^2+2*a*x+a^2)/4];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[clairault,0,0];

/* 42 */
(pn_(42),ans:contrib_ode(eqn:'diff(y,x)^2-2*x*'diff(y,x)+1=0,y,x));
[-(2*y-log(2*sqrt(x^2-1)+2*x)+x*sqrt(x^2-1)-x^2)/2=%c];
solve(ans[1],y);
[y=(log(2*sqrt(x^2-1)+2*x)-x*sqrt(x^2-1)+x^2-2*%c)/2];
[method,ode_check(eqn,%)];
[lie,0];

/* 43 */
(pn_(43),ans:contrib_ode(eqn:'diff(y,x)^2+2*x*'diff(y,x)-3*x^2=0,y,x));
[y = x^2/2+%c,y = %c-3*x^2/2];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[factor,0,0];

/* 44 */
(pn_(44),ans:contrib_ode(eqn:'diff(y,x)^2*x*'diff(y,x)-y=0,y,x));
[[x = %c*%e^-(3*log(%t^2-1)/2),y = %t^3*x]]; /* parametric soln */
first(solve(ans[1][1]^2,%t));
%t=-sqrt(x^(2/3)+%c^(2/3))/x^(1/3);
ans:ev(ans[1][2],%);
y=-(x^(2/3)+%c^(2/3))^(3/2);
[method,ode_check(eqn,ans)];
[lagrange,0];

/* 45 */
(pn_(45),ans:contrib_ode(eqn:'diff(y,x)^2+2*x*diff(y,x)-y=0,y,x));
[4*y-x^2+2*%c*x-%c^2=0];
ans:solve(ans[1],y);
[y = (x^2-2*%c*x+%c^2)/4];
[method,ode_check(eqn,ans[1])];
[lagrange,0];

/* 46 */
(pn_(46),ans:contrib_ode(eqn:'diff(y,x)^2-2*x*'diff(y,x)+2*y=0,y,x));
[y=(2*%c*x-%c^2)/2,y=x^2/2];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[clairault,0,0];

/* 47 CHECKME */
(pn_(47),ans:contrib_ode(eqn:'diff(y,x)^2+(1+2*x)*diff(y,x)-x*(1-x)=0,y,x));
[-(8*y+asin(2*x-1)+(4*x-2)*sqrt(x-x^2))/8 = %c];
solve(ans[1],y);
[y = -(asin(2*x-1)+(4*x-2)*sqrt(x-x^2)+8*%c)/8];
[method,ode_check(eqn,%)];
[lie,-(16*x^4-32*x^3+sqrt(4*x-4*x^2)*sqrt(x-x^2)*(8*x^2-8*x+1)+18*x^2-2*x)/(64*x^4-128*x^3+64*x^2)];

/* 48 */
(pn_(48),ans:contrib_ode(eqn:'diff(y,x)^2+(1-x)*diff(y,x)-2*(x-y)=0,y,x));
[-(2*sqrt(x-y)-sqrt(2)*log(sqrt(2)*sqrt(x-y)+1)-sqrt(2)*x)/sqrt(2)=%c];
method;
lie;

/* 49 */
(pn_(49),ans:contrib_ode(eqn:'diff(y,x)^2+3*x*'diff(y,x)-y=0,y,x));
[[x = %e^-(3*log(%t)/2)*(%c-2*%e^(5*log(%t)/2)/5),y = 3*%t*x+%t^2]]; /* CHECKME */
method;
lagrange;

/* 50 */
(pn_(50),ans:contrib_ode(eqn:'diff(y,x)^2-4*(1+x)*'diff(y,x)+4*y=0,y,x));
[y=(4*%c*x-%c^2+4*%c)/4,y=x^2+2*x+1];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[clairault,0,0];

/* 51 */
assume(4*c+a^2>0,b>0);
[4*c+a^2 > 0,b > 0];
(pn_(51),ans:contrib_ode(eqn:'diff(y,x)^2+a*x*'diff(y,x)=b+c*x^2,y,x));
[-(sqrt(4*c+a^2)*(4*y+a*x^2)+4*b*asinh(sqrt(4*c+a^2)*x/(2*sqrt(b)))+sqrt(4*c+a^2)*x*sqrt((4*c+a^2)*x^2+4*b))/(4*sqrt(4*c+a^2))=%c];
solve(ans[1],y);
[y = -(4*b*asinh(sqrt(4*c+a^2)*x/(2*sqrt(b)))+sqrt(4*c+a^2)*x*sqrt(4*c*x^2+a^2*x^2+4*b)+a*sqrt(4*c+a^2)*x^2+4*%c*sqrt(4*c+a^2))/(4*sqrt(4*c+a^2))];
[method,ode_check(eqn,%[1])];
[lie,0];
forget(4*c+a^2>0,b>0);
[4*c+a^2>0,b>0];

/* 52 */
(pn_(52),ans:contrib_ode(eqn:'diff(y,x)^2-a*x*'diff(y,x)+a*y=0,y,x));
[y=(%c*a*x-%c^2)/a,y=a*x^2/4];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[clairault,0,0];

/* 53 ode1_lie takes forever */
/* (pn_(53),ans:contrib_ode(eqn:'diff(y,x)^2+a*x*'diff(y,x)+b*x^2+c*y=0,y,x));*/

/* 54 */
(pn_(54),ans:contrib_ode(eqn:'diff(y,x)^2+(a+b*x)*'diff(y,x)+c=b*y,y,x));
[y=(%c*b*x+c+%c*a+%c^2)/b,y=-(b^2*x^2+2*a*b*x-4*c+a^2)/(4*b)];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[clairault,0,0];

/* 55 ode1_lie takes forever */
/*(pn_(55),ans:contrib_ode(eqn:'diff(y,x)^2-2*x^2*'diff(y,x)+2*x*y=0,y,x));*/

/* 56 ode1_lie takes forever */
/*(pn_(56),ans:contrib_ode(eqn:'diff(y,x)^2-a*x^2*'diff(y,x)+b*x*y=0,y,x));*/

/* 57 */
(pn_(57),ans:contrib_ode(eqn:'diff(y,x)^2-a*x^3*'diff(y,x)+2*a*x^2*y=0,y,x));
[-(sqrt(a)*sqrt(a^2*x^4-8*a*y)-a^(3/2)*x^2)/(sqrt(3)*a) = %c];
solve(ans[1],sqrt(a^2*x^4-8*a*y));
[sqrt(a^2*x^4-8*a*y) = (a^(3/2)*x^2-sqrt(3)*%c*a)/sqrt(a)];
%[1]^2;
a^2*x^4-8*a*y = (a^(3/2)*x^2-sqrt(3)*%c*a)^2/a;
ans:solve(%,y);
[y=(2*sqrt(3)*%c*a^(5/2)*x^2-3*%c^2*a^2)/(8*a^2)];
[method,ode_check(eqn,ans[1])];
[lie,0];

/* 58 ode1_lie takes forever */
/* (pn_(58),ans:contrib_ode(eqn:'diff(y,x)^2-a*x^3*'diff(y,x)+4*a*x^2*y=0,y,x));*/

/* 59 */
(pn_(59),ans:contrib_ode(eqn:'diff(y,x)^2+4*x^5*'diff(y,x)-12*x^4*y=0,y,x));
[(4*%i*sqrt(3*y+x^6)+4*%i*x^3)/(3*sqrt(2)) = %c];
ans[1]/%i,%c=%i*%c;
-%i*(4*%i*sqrt(3*y+x^6)+4*%i*x^3)/(3*sqrt(2)) = %c;
expand(%);
4*sqrt(3*y+x^6)/(3*sqrt(2))+4*x^3/(3*sqrt(2)) = %c;
solve(%,sqrt(3*y+x^6));
[sqrt(3*y+x^6) = -(4*x^3-3*sqrt(2)*%c)/4];
%[1]^2;
3*y+x^6 = (4*x^3-3*sqrt(2)*%c)^2/16;
ans:solve(%,y);
[y = -(4*sqrt(2)*%c*x^3-3*%c^2)/8];
[method,ode_check(eqn,ans[1])];
[lie,0];

/* 60 */
(pn_(60),ans:contrib_ode(eqn:'diff(y,x)^2+2*'diff(y,x)*cosh(x)+1=0,y,x));
[-%e^-x*(2*%e^x*y+2*%e^x*sinh(x)+%e^(2*x)+1)/2 = %c];
solve(ans[1],y);
[y = -%e^-x*(2*%e^x*sinh(x)+%e^(2*x)+2*%c*%e^x+1)/2];
expand(ev(%[1],sinh(x)=1/2*(exp(x)-exp(-x))));
y=-%e^x-%c;
ode_check(eqn,%);
-2*%e^x*cosh(x)+%e^(2*x)+1;
expand(ev(%,cosh(x)=1/2*(exp(x)+exp(-x))));
0;
method;
lie;

/* 61 */
(pn_(61),ans:contrib_ode(eqn:'diff(y,x)^2+y*'diff(y,x)=x*(x+y),y,x));
[y = x^2/2+%c,y = %e^-x*(%c-(x-1)*%e^x)];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
[factor,0,0];

/* 62 */
(pn_(62),ans:contrib_ode(eqn:'diff(y,x)^2-y*'diff(y,x)+exp(x)=0,y,x));
[2*log(sqrt(y^2-4*%e^x)+y)=%c];
map(ex...
 
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