87.6.9 problem 9

Internal problem ID [23326]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 53
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:37:15 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 19

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

\[ y = \sqrt {-\frac {1}{\operatorname {LambertW}\left (-c_1 \,x^{2}\right )}}\, x \]

✓ Mathematica. Time used: 5.733 (sec). Leaf size: 56

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

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

✓ Sympy. Time used: 0.771 (sec). Leaf size: 19

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

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