Internal problem ID [5003]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]
Solve \begin {gather*} \boxed {x y^{\prime }-y-\left (x +y\right ) \ln \left (\frac {x +y}{x}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 14
dsolve(x*diff(y(x),x)-y(x)=(x+y(x))*ln((x+y(x))/x),y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{c_{1} x} x -x \]
✓ Solution by Mathematica
Time used: 0.562 (sec). Leaf size: 24
DSolve[x*y'[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*}