29.25.11 problem 708

Internal problem ID [5298]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 25
Problem number : 708
Date solved : Sunday, March 30, 2025 at 07:52:01 AM
CAS classification : [_rational]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 197

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

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

✓ Mathematica. Time used: 6.521 (sec). Leaf size: 272

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

\begin{align*} y(x)\to -\frac {\sqrt {-\sqrt {\left (a^2+x^2-c_1{}^2\right ){}^2+4 c_1{}^2 x^2}-a^2-x^2+c_1{}^2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-\sqrt {\left (a^2+x^2-c_1{}^2\right ){}^2+4 c_1{}^2 x^2}-a^2-x^2+c_1{}^2}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {\sqrt {\left (a^2+x^2-c_1{}^2\right ){}^2+4 c_1{}^2 x^2}-a^2-x^2+c_1{}^2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {\sqrt {\left (a^2+x^2-c_1{}^2\right ){}^2+4 c_1{}^2 x^2}-a^2-x^2+c_1{}^2}}{\sqrt {2}} \\ y(x)\to 0 \\ y(x)\to -\sqrt {-x^2} \\ y(x)\to \sqrt {-x^2} \\ \end{align*}

✗ Sympy

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

Timed Out