23.1.529 problem 519

Internal problem ID [5136]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 519
Date solved : Tuesday, September 30, 2025 at 11:44:01 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x y y^{\prime }+x^{2} {\mathrm e}^{-\frac {2 y}{x}}-y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 18

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

\[ y = \frac {\left (\operatorname {LambertW}\left (-4 \left (\ln \left (x \right )+c_1 \right ) {\mathrm e}^{-1}\right )+1\right ) x}{2} \]

✓ Mathematica. Time used: 60.103 (sec). Leaf size: 25

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

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

✗ Sympy

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

TypeError : cannot determine truth value of Relational: -1 < 4*x