23.1.523 problem 513

Internal problem ID [5130]
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 : 513
Date solved : Tuesday, September 30, 2025 at 11:43:40 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

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

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

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

✓ Mathematica. Time used: 1.081 (sec). Leaf size: 25

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

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

✗ Sympy

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

RecursionError : maximum recursion depth exceeded