47.2.8 problem 8

Internal problem ID [7424]
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 : Monday, January 27, 2025 at 02:54:47 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

Time used: 0.453 (sec). Leaf size: 24

DSolve[x*D[y[x],x]-y[x]==(x+y[x])*Log[ (x+y[x])/x],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*}