23.2.231 problem 237

Internal problem ID [5586]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 237
Date solved : Tuesday, September 30, 2025 at 01:02:09 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} \left (a^{2} x^{2}-y^{2}\right ) {y^{\prime }}^{2}-2 x y y^{\prime }+\left (a^{2}-1\right ) x^{2}&=0 \end{align*}

✓ Maple. Time used: 0.070 (sec). Leaf size: 157

ode:=(a^2*x^2-y(x)^2)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+(a^2-1)*x^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {a^{2}-1}\, x \\ y &= -\sqrt {a^{2}-1}\, x \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{3}-a^{2} \textit {\_a} -\sqrt {a^{2} \left (\textit {\_a}^{2}-a^{2}+1\right )}+\textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2}-a^{2}+1\right )}d \textit {\_a} +c_1 \right ) x \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}-a^{2} \textit {\_a} +\sqrt {a^{2} \left (\textit {\_a}^{2}-a^{2}+1\right )}+\textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2}-a^{2}+1\right )}d \textit {\_a} +c_1 \right ) x \\ \end{align*}

✓ Mathematica. Time used: 23.064 (sec). Leaf size: 1295

ode=(a^2+x^2-y[x]^2)*D[y[x],x]^2-2*x*y[x]*D[y[x],x]+(a^2-1)*x^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(x**2*(a**2 - 1) - 2*x*y(x)*Derivative(y(x), x) + (a**2*x**2 - y(x)**2)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out