41.2.22 problem 22

Internal problem ID [8724]
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 : 22
Date solved : Tuesday, September 30, 2025 at 05:44:18 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

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

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

\[ y = {\mathrm e}^{c_1 x +1} x \]

✓ Mathematica. Time used: 0.126 (sec). Leaf size: 24

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

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

✓ Sympy. Time used: 0.451 (sec). Leaf size: 10

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

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