41.2.8 problem 8

Internal problem ID [8710]
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 : 8
Date solved : Tuesday, September 30, 2025 at 05:43:05 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 12

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

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

✓ Mathematica. Time used: 0.24 (sec). Leaf size: 24

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

\begin{align*} y(x)&\to x \left (-1+e^{e^{-c_1} x}\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.575 (sec). Leaf size: 10

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

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