2.14 problem 14

Internal problem ID [5009]

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: 14.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [[_homogeneous, class A], _rational, [_Abel, 2nd type, class A]]

Solve \begin {gather*} \boxed {x y^{\prime }-y-y^{\prime } y=0} \end {gather*}

Solution by Maple

Time used: 0.016 (sec). Leaf size: 15

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

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

Solution by Mathematica

Time used: 60.03 (sec). Leaf size: 20

DSolve[x*y'[x]-y[x]==y[x]*y'[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to e^{\text {ProductLog}\left (-e^{-c_1} x\right )+c_1} \\ \end{align*}