23.4.269 problem 269

Internal problem ID [6571]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 269
Date solved : Friday, October 03, 2025 at 02:09:32 AM
CAS classification : [NONE]

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 111

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

\begin{align*} y &= \operatorname {RootOf}\left (1+\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {c_1 \,\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}+2 c_1 \,\textit {\_f}^{2}+a^{2}+c_1}}{c_1 \,\textit {\_f}^{2}+a^{2}+c_1}d \textit {\_f} +c_2 \right ) x \right ) x \\ y &= \operatorname {RootOf}\left (1+\left (-\int _{}^{\textit {\_Z}}\frac {\sqrt {c_1 \,\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}+2 c_1 \,\textit {\_f}^{2}+a^{2}+c_1}}{c_1 \,\textit {\_f}^{2}+a^{2}+c_1}d \textit {\_f} +c_2 \right ) x \right ) x \\ \end{align*}

✗ Mathematica

ode=a^2*y[x] + (x^2 + y[x]^2)^2*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(a**2*y(x) + (x**2 + y(x)**2)**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve a**2*y(x) + (x**2 + y(x)**2)**2*Derivative(y(x), (x, 2))