69.1.4 problem 4

Internal problem ID [14086]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 4
Date solved : Monday, March 31, 2025 at 12:03:07 PM
CAS classification : [_rational]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.354 (sec). Leaf size: 75

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

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

✗ Sympy

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

Timed Out