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(load("contrib_ode"),0);
0$
/* ODE tests - Second order linear equations 401-445
Reference:
E Kamke, Differentialgleichungen Losungsmethoden und Losungen,
Vol 1, Geest & Portig, Leipzig, 1961
*/
/* Print ode number*/
(pn_(n_):=print("Kamke ODE 2.",n_),true);
true;
/* Kamke 2. 401 */
assume(b>0,4*b>a^2);
[b>0,4*b>a^2];
(pn_(401),ans:contrib_ode(eqn:'diff(y,x,2)=-(3*x^2+a)*'diff(y,x,1)/x^3-b*y/x^6,y,x));
[y=(%k2*cos(sqrt(4-a^2/b)*sqrt(b)/(4*x^2))-%k1*sin(sqrt(4-a^2/b)*sqrt(b)/(4*x^2)))*%e^(a/(4*x^2))];
ode_check(eqn,ans[1]);
0;
method;
xformtoconstcoeff;
forget(b>0,4*b>a^2);
[b>0,4*b>a^2];
/* Kamke 2. 402 */
(pn_(402),ans:contrib_ode(eqn:'diff(y,x,2)=-((1-4*a)*x^2-1)*'diff(y,x,1)/(x*(x^2-1))-(4*a*(a+1)*x^4+(x^2-1)^2*(x^2-v^2)-2*a*x^2*(x^2-1))*y/(x^2*(x^2-1)^2),y,x));
false;
/* Kamke 2. 403 */
/* This equation presents a problem - runs forever */
/*(pn_(403),ans:contrib_ode(eqn:'diff(y,x,2)=(-(-b2-a2+1)/(x-c_2)-(-b1-a1+1)/(x-c_1)-(-b3-a3+1)/(x-c3))*'diff(y,x,1)-(a2*b2*(c_2-c3)*(c_2-c_1)/(x-c_2)+a1*b1*(c_1-c3)*(c_1-c_2)/(x-c_1)+a3*b3*(c3-c_1)*(c3-c_2)/(x-c3))*y/((x-c3)*(x-c_1)*(x-c_2)),y,x));*/
/* Kamke 2. 404 */
(pn_(404),ans:contrib_ode(eqn:'diff(y,x,2)=(-2*x^2-1)*'diff(y,x,1)/x^3-(1-2*x^2)*y/(4*x^6),y,x));
[y = %k2*%e^(1/(4*x^2))/x+%k1*%e^(1/(4*x^2))];
[method,ode_check(eqn,ans[1])];
[odelin,0];
/* Kamke 2. 405 */
assume(notequal(a,9));
[notequal(a,9)];
(pn_(405),ans:contrib_ode(eqn:'diff(y,x,2)=(2*x^2+1)*'diff(y,x,1)/x^3-(a*x^4+10*x^2+1)*y/(4*x^6),y,x));
[y = %k2*x^(sqrt(9-a)/2+3/2)*%e^-(1/(4*x^2))
+%k1*x^(3/2-sqrt(9-a)/2)*%e^-(1/(4*x^2))];
[method,ode_check(eqn,ans[1])];
[odelin,0];
forget(notequal(a,9));
[notequal(a,9)];
/* Kamke 2. 406 */
(pn_(406),ans:contrib_ode(eqn:'diff(y,x,2)=-27*x*y/(16*(x^3-1)^2),y,x));
false;
/* Kamke 2. 407 */
/* This equation presents a problem - runs forever */
/*(pn_(407),ans:contrib_ode(eqn:'diff(y,x,2)=(-b3*(-bl3-al3+1)/(b3*x-a3)-b2*(-bl2-al2+1)/(b2*x-a2)-b1*(-bl1-al1+1)/(b1*x-a1))*'diff(y,x,1)-(al3*(a3*b1-a1*b3)*(a2*b3-a3*b2)*bl3/(b3*x-a3)+al2*(a1*b2-a2*b1)*(a2*b3-a3*b2)*bl2/(b2*x-a2)+al1*(a1*b2-a2*b1)*(a3*b1-a1*b3)*bl1/(b1*x-a1))*y/((b1*x-a1)*(b2*x-a2)*(b3*x-a3)),y,x));*/
/* Kamke 2. 408 */
/* CHECKME: Takes a long time
(pn_(408),ans:contrib_ode(eqn:'diff(y,x,2)=((x^2-a1)*(x^2-a2)*(x^2-a3)-x^2*((x^2-a2)*(x^2-a3)+(x^2-a1)*(x^2-a3)+(x^2-a1)*(x^2-a2)))*'diff(y,x,1)/(x*(x^2-a1)*(x^2-a2)*(x^2-a3))-y*(B+x^2*A)/(x*(x^2-a1)*(x^2-a2)*(x^2-a3)),y,x));
false;
*/
/* Kamke 2. 409 */
assume(a>1);
[a>1];
(pn_(409),ans:contrib_ode(eqn:'diff(y,x,2)=-a*'diff(y,x,1)/x-b^2*y/x^(2*a),y,x));
[y=%k1*sin(b*x^(1-a)/(1-a))+%k2*cos(b*x^(1-a)/(1-a))];
ode_check(eqn,ans[1]);
0;
method;
xformtoconstcoeff;
forget(a>1);
[a>1];
/* Kamke 2. 410 */
(pn_(410),ans:contrib_ode(eqn:'diff(y,x,2)=(-a*p*x^b-q)*'diff(y,x,1)/(x*(a*x^b-1))-(a*r*x^b+s)*y/(x^2*(a*x^b-1)),y,x));
false;
/* Kamke 2. 411 */
(pn_(411),ans:contrib_ode(eqn:'diff(y,x,2)=y/(%e^x+1),y,x));
false;
/* Kamke 2. 412 */
(pn_(412),ans:contrib_ode(eqn:'diff(y,x,2)='diff(y,x,1)/(x*log(x))+log(x)^2*y,y,x));
[y=%i*%k1*sinh(x*log(x)-x)+%k2*cosh(x*log(x)-x)];
ode_check(eqn,ans[1]);
0;
method;
xformtoconstcoeff;
/* Kamke 2. 413 */
(pn_(413),ans:contrib_ode(eqn:'diff(y,x,2)='diff(y,x,1)/(x*(log(x)-1))-y/(x^2*(log(x)-1)),y,x));
false;
/* Kamke 2. 414 */
(pn_(414),ans:contrib_ode(eqn:'diff(y,x,2)=(a^2*sinh(x)^2+(n-1)*n)*y/sinh(x)^2,y,x));
false;
/* Kamke 2. 415 */
(pn_(415),ans:contrib_ode(eqn:'diff(y,x,2)=-2*n*cosh(x)*'diff(y,x,1)/sinh(x)-(n^2-a^2)*y,y,x));
false;
/* Kamke 2. 416 */
(pn_(416),ans:contrib_ode(eqn:'diff(y,x,2)=(-2*n-1)*cos(x)*'diff(y,x,1)/sin(x)-(v-n)*(v+n+1)*y,y,x));
false;
/* Kamke 2. 417 */
(pn_(417),ans:contrib_ode(eqn:'diff(y,x,2)=(cos(x)-sin(x)^2)*'diff(y,x,1)/sin(x)-sin(x)^2*y,y,x));
[y=%e^(cos(x)/2)*(%k2*cos(sqrt(3)*cos(x)/2)-%k1*sin(sqrt(3)*cos(x)/2))];
ode_check(eqn,ans[1]);
0;
method;
xformtoconstcoeff;
/* Kamke 2. 418 */
(pn_(418),ans:contrib_ode(eqn:'diff(y,x,2)=sin(x)*y/(x*cos(x)-sin(x))-x*sin(x)*'diff(y,x,1)/(x*cos(x)-sin(x)),y,x));
false;
/* Kamke 2. 419 */
(pn_(419),ans:contrib_ode(eqn:'diff(y,x,2)=(2*x*cos(x)-x^2*sin(x))*'diff(y,x,1)/(x^2*cos(x))-(2*x*cos(x)-x*sin(x))*y/(x^2*cos(x)),y,x));
false;
/* Kamke 2. 420 */
(pn_(420),ans:contrib_ode(eqn:cos(x)^2*'diff(y,x,2)-(a*cos(x)^2+(n-1)*n)*y,y,x));
false;
/* Kamke 2. 421 */
(pn_(421),ans:contrib_ode(eqn:'diff(y,x,2)=-a*(n-1)*sin(2*a*x)*'diff(y,x,1)/cos(a*x)^2-a^2*n*((n-1)*sin(a*x)^2+cos(a*x)^2)*y/cos(a*x)^2,y,x));
false;
/* Kamke 2. 422 */
(pn_(422),ans:contrib_ode(eqn:'diff(y,x,2)=2*y/sin(x)^2,y,x));
false;
/* Kamke 2. 423 */
(pn_(423),ans:contrib_ode(eqn:'diff(y,x,2)=-a*y/sin(x)^2,y,x));
false;
/* Kamke 2. 424 */
(pn_(424),ans:contrib_ode(eqn:sin(x)^2*'diff(y,x,2)-(a*sin(x)^2+(n-1)*n)*y,y,x));
false;
/* Kamke 2. 425 */
(pn_(425),ans:contrib_ode(eqn:'diff(y,x,2)=(a^2*cos(x)^2+(3-2*a)*cos(x)-3*a+3)*y/sin(x)^2,y,x));
false;
/* Kamke 2. 426 */
(pn_(426),ans:contrib_ode(eqn:sin(x)^2*'diff(y,x,2)-(a^2*cos(x)^2+b*cos(x)+b^2/(2*a-3)^2+3*a+2)*y,y,x));
false;
/* Kamke 2. 427 */
(pn_(427),ans:contrib_ode(eqn:'diff(y,x,2)=(a*(a+1)*b*sin(2*x)-((a+1)^2-a^2*b^2)*sin(x)^2+(a-1)*a)*y/sin(x)^2,y,x));
false;
/* Kamke 2. 428 */
(pn_(428),ans:contrib_ode(eqn:'diff(y,x,2)=(-b*sin(x)^2-a*cos(x)^2-c)*y/sin(x)^2,y,x));
false;
/* Kamke 2. 429 */
(pn_(429),ans:contrib_ode(eqn:'diff(y,x,2)=y/sin(x)^2-cos(x)*'diff(y,x,1)/sin(x),y,x));
[y=%k2*cosh(log(cos(x)+1)/2-log(cos(x)-1)/2)-%i*%k1*sinh(log(cos(x)+1)/2-log(cos(x)-1)/2)];
[method,ode_check(eqn,ans[1])];
[xformtoconstcoeff,0];
/* Kamke 2. 430 */
(pn_(430),ans:contrib_ode(eqn:'diff(y,x,2)=-cos(x)*'diff(y,x,1)/sin(x)-(v*(v+1)*sin(x)^2-n^2)*y/sin(x)^2,y,x));
false;
/* Kamke 2. 431 */
(pn_(431),ans:contrib_ode(eqn:'diff(y,x,2)=cos(2*x)*'diff(y,x,1)/sin(2*x)-2*y,y,x));
false;
/* Kamke 2. 432 */
(pn_(432),ans:contrib_ode(eqn:'diff(y,x,2)=-cos(x)*'diff(y,x,1)/sin(x)-(-17*sin(x)^2-1)*y/(4*sin(x)^2),y,x));
false;
/* Kamke 2. 433 */
(pn_(433),ans:contrib_ode(eqn:'diff(y,x,2)=-sin(x)*'diff(y,x,1)/cos(x)-(x^2*sin(x)^2-24*cos(x)^2+2*x^2)*y/(4*x^2*cos(x)^2)+sqrt(cos(x)),y,x));
false;
/* Kamke 2. 434 */
(pn_(434),ans:contrib_ode(eqn:'diff(y,x,2)=-b*cos(x)*'diff(y,x,1)/(a*sin(x))-(c*cos(x)^2+d*cos(x)+e)*y/(a*sin(x)^2),y,x));
false;
/* Kamke 2. 435 */
(pn_(435),ans:contrib_ode(eqn:'diff(y,x,2)=-4*sin(3*x)*y/sin(x)^3,y,x));
false;
/* Kamke 2. 436 */
(pn_(436),ans:contrib_ode(eqn:'diff(y,x,2)=-(4*v*(v+1)*sin(x)^2-cos(x)^2-4*n^2+2)*y/(4*sin(x)^2),y,x));
false;
/* Kamke 2. 437 */
(pn_(437),ans:contrib_ode(eqn:'diff(y,x,2)=(3*sin(x)^2+1)*'diff(y,x,1)/(cos(x)*sin(x))+sin(x)^2*y/cos(x)^2,y,x));
false;
/* Kamke 2. 438 */
(pn_(438),ans:contrib_ode(eqn:'diff(y,x,2)=(a*cos(x)^2*sin(x)^2+(m-1)*m*sin(x)^2+(n-1)*n*cos(x)^2)*y/(cos(x)^2*sin(x)^2),y,x));
false;
/* Kamke 2. 439 */
/* This equation presents a problem - crashes ode_contrib */
/*(pn_(439),ans:contrib_ode(eqn:ode[439],y,x));*/
/* Kamke 2. 440 */
(pn_(440),ans:contrib_ode(eqn:'diff(y,x,2)=('diff(phi(x),x,2)+phi(x)*'diff(phi(x),x,1)-phi(x^3))*'diff(y,x,1)/('diff(phi(x),x,1)+phi(x)^2)-(-phi(x)*'diff(phi(x),x,2)+('diff(phi(x),x,1))^2-phi(x)^2*'diff(phi(x),x,1))*y/('diff(phi(x),x,1)+phi(x)^2),y,x));
false;
/* Kamke 2. 441 */
(pn_(441),ans:contrib_ode(eqn:'diff(y,x,2)=(2*jacobi_sn(x,m)+jacobi_cn(x,m)*jacobi_dn(x,m))*'diff(y,x,1)/(jacobi_sn(x,m)^2-jacobi_sn(a,m)^2)-(-4*jacobi_sn(a,m)^2*(m+1)+6*jacobi_sn(a,m)^4*m+2)*y/(jacobi_sn(x,m)^2-jacobi_sn(a,m)^2)-1/(jacobi_sn(x,m)^2-jacobi_sn(a,m)^2),y,x));
false;
/* Kamke 2. 442 */
(pn_(442),ans:contrib_ode(eqn:'diff(y,x,2)=y/f(x)-x*'diff(y,x,1)/f(x),y,x));
false;
/* Kamke 2. 443 */
(pn_(443),ans:contrib_ode(eqn:'diff(y,x,2)=-'diff(f(x),x,1)*'diff(y,x,1)/(2*f(x))-g(x)*y/f(x),y,x));
false;
/* Kamke 2. 444 */
(pn_(444),ans:contrib_ode(eqn:'diff(y,x,2)=-a*'diff(f(x),x,1)*'diff(y,x,1)/f(x)-b*f(x)^(2*a)*y,y,x));
false;
/* Kamke 2. 445 */
(pn_(445),ans:contrib_ode(eqn:'diff(y,x,2)=((g(x)^2-1)*(f(x)*'diff(g(x),x,2)+2*'diff(f(x),x,1)*'diff(g(x),x,1))-2*f(x)*g(x)*('diff(g(x),x,1))^2)*'diff(y,x,1)/(f(x)*(g(x)^2-1)*'diff(g(x),x,1))-((g(x)^2-1)*('diff(f(x),x,1)*(f(x)*'diff(g(x),x,2)+2*'diff(f(x),x,1)*'diff(g(x),x,1))-f(x)*'diff(f(x),x,2)*'diff(g(x),x,1))-f(x)*('diff(g(x),x,1))^2*(v*(v+1)*f(x)*'diff(g(x),x,1)+2*g(x)*'diff(f(x),x,1)))*y/(f(x)^2*(g(x)^2-1)*'diff(g(x),x,1)),y,x));
false;
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