1.5.5 problem 5

Internal problem ID [109]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.6 (substitution and exact equations). Problems at page 72
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 03:43:43 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 3.052 (sec). Leaf size: 25

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

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

✓ Sympy. Time used: 0.706 (sec). Leaf size: 15

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

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