41.2.29 problem 29

Internal problem ID [8731]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 29
Date solved : Tuesday, September 30, 2025 at 05:45:54 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 13

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

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

✓ Mathematica. Time used: 2.6 (sec). Leaf size: 23

ode=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\right )}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.317 (sec). Leaf size: 12

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

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