47.2.22 problem 22

Internal problem ID [7438]
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 : Monday, January 27, 2025 at 02:59:27 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 12

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

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

✓ Solution by Mathematica

Time used: 0.223 (sec). Leaf size: 24

DSolve[x*D[y[x],x]==y[x]*Log[y[x]/x],y[x],x,IncludeSingularSolutions -> True]
 

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