23.1.688 problem 683

Internal problem ID [5295]
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 : 683
Date solved : Tuesday, September 30, 2025 at 12:15:23 PM
CAS classification : [_exact, _rational]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 113

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

\begin{align*} y &= \sqrt {-x^{2}-a -2 \sqrt {a \,x^{2}-c_1}} \\ y &= \sqrt {-x^{2}-a +2 \sqrt {a \,x^{2}-c_1}} \\ y &= -\sqrt {-x^{2}-a -2 \sqrt {a \,x^{2}-c_1}} \\ y &= -\sqrt {-x^{2}-a +2 \sqrt {a \,x^{2}-c_1}} \\ \end{align*}

✓ Mathematica. Time used: 6.228 (sec). Leaf size: 149

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

\begin{align*} y(x)&\to -\sqrt {-\sqrt {a^2+4 a x^2+4 c_1}-a-x^2}\\ y(x)&\to \sqrt {-\sqrt {a^2+4 a x^2+4 c_1}-a-x^2}\\ y(x)&\to -\sqrt {\sqrt {a^2+4 a x^2+4 c_1}-a-x^2}\\ y(x)&\to \sqrt {\sqrt {a^2+4 a x^2+4 c_1}-a-x^2} \end{align*}

✗ Sympy

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

Timed Out