81.3.13 problem 4-18

Internal problem ID [21512]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 4. Homogeneous differential equations.
Problem number : 4-18
Date solved : Thursday, October 02, 2025 at 07:46:20 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y^{\prime }&=\frac {2 x y}{x^{2}-y^{2}} \end{align*}

✓ Maple. Time used: 0.057 (sec). Leaf size: 47

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

\begin{align*} y &= \frac {1-\sqrt {-4 c_1^{2} x^{2}+1}}{2 c_1} \\ y &= \frac {1+\sqrt {-4 c_1^{2} x^{2}+1}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.585 (sec). Leaf size: 66

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

\begin{align*} y(x)&\to \frac {1}{2} \left (e^{c_1}-\sqrt {-4 x^2+e^{2 c_1}}\right )\\ y(x)&\to \frac {1}{2} \left (\sqrt {-4 x^2+e^{2 c_1}}+e^{c_1}\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.834 (sec). Leaf size: 44

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {- 4 x^{2} + e^{2 C_{1}}}}{2} + \frac {e^{C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {- 4 x^{2} + e^{2 C_{1}}}}{2} + \frac {e^{C_{1}}}{2}\right ] \]