23.1.687 problem 682

Internal problem ID [5294]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 682
Date solved : Tuesday, September 30, 2025 at 12:15:21 PM
CAS classification : [_rational]

\begin{align*} \left (a^{2} x +y \left (x^{2}-y^{2}\right )\right ) y^{\prime }+x \left (x^{2}-y^{2}\right )&=a^{2} y \end{align*}

✗ Maple

ode:=(a^2*x+(x^2-y(x)^2)*y(x))*diff(y(x),x)+x*(x^2-y(x)^2) = a^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.191 (sec). Leaf size: 126

ode=(a^2*x+y[x]*(x^2-y[x]^2))*D[y[x],x]+x*(x^2-y[x]^2)==a^2*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x\left (-\frac {a^2}{2 (K[1]-y(x))}+\frac {a^2}{2 (K[1]+y(x))}+K[1]\right )dK[1]+\int _1^{y(x)}\left (-\frac {a^2}{2 (K[2]-x)}+\frac {a^2}{2 (x+K[2])}+K[2]-\int _1^x\left (-\frac {a^2}{2 (K[1]-K[2])^2}-\frac {a^2}{2 (K[1]+K[2])^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**2*y(x) + x*(x**2 - y(x)**2) + (a**2*x + (x**2 - y(x)**2)*y(x))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out