54.1.481 problem 494

Internal problem ID [11795]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 494
Date solved : Tuesday, September 30, 2025 at 10:37:16 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.133 (sec). Leaf size: 157

ode:=(y(x)^2-a^2*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: 0.177 (sec). Leaf size: 80

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

\begin{align*} y(x)&\to a c_1-\sqrt {-x^2+c_1{}^2}\\ y(x)&\to a c_1+\sqrt {-x^2+c_1{}^2}\\ y(x)&\to -\sqrt {a^2-1} x\\ y(x)&\to \sqrt {a^2-1} x \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(x**2*(1 - a**2) + 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