54.1.494 problem 507

Internal problem ID [11808]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 507
Date solved : Sunday, October 12, 2025 at 01:58:11 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} \left (y^{4}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 a^{2} x y y^{\prime }+y^{2} \left (y^{2}-a^{2}\right )&=0 \end{align*}

✓ Maple. Time used: 1.293 (sec). Leaf size: 207

ode:=(y(x)^4-a^2*x^2)*diff(y(x),x)^2+2*a^2*x*y(x)*diff(y(x),x)+y(x)^2*(y(x)^2-a^2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y-\operatorname {RootOf}\left (\sqrt {\operatorname {RootOf}\left (\left (-y^{4}+a^{2} x^{2}\right ) \textit {\_Z}^{2}-y^{2}+a^{2}-2 \textit {\_Z} \,a^{2} x \right ) \textit {\_Z}}\, c_1 +a \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {3}{4}\right ], \frac {\textit {\_Z}^{2} \left (2 \operatorname {RootOf}\left (\left (-y^{4}+a^{2} x^{2}\right ) \textit {\_Z}^{2}-y^{2}+a^{2}-2 \textit {\_Z} \,a^{2} x \right ) a^{2} x +\textit {\_Z}^{2}-a^{2}\right )}{\textit {\_Z}^{4}-a^{2} x^{2}}\right )+\textit {\_Z} \left (-\frac {a^{2} \left (2 \operatorname {RootOf}\left (\left (-y^{4}+a^{2} x^{2}\right ) \textit {\_Z}^{2}-y^{2}+a^{2}-2 \textit {\_Z} \,a^{2} x \right ) \textit {\_Z}^{2} x -\textit {\_Z}^{2}+x^{2}\right )}{\textit {\_Z}^{4}-a^{2} x^{2}}\right )^{{1}/{4}}\right ) &= 0 \\ \end{align*}

✗ Mathematica

ode=y[x]^2*(-a^2 + y[x]^2) + 2*a^2*x*y[x]*D[y[x],x] + (-(a^2*x^2) + y[x]^4)*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(2*a**2*x*y(x)*Derivative(y(x), x) + (-a**2 + y(x)**2)*y(x)**2 + (-a**2*x**2 + y(x)**4)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out