Internal problem ID [12121]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 1, First-Order Differential Equations. Problems page 88
Problem number: Problem 10.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _dAlembert]
\[ \boxed {x \left (\ln \left (x \right )-\ln \left (y\right )\right ) y^{\prime }-y=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 14
dsolve(x*(ln(x)-ln(y(x)))*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\operatorname {LambertW}\left (c_{1} x \,{\mathrm e}^{-1}\right )}{c_{1}} \]
✓ Solution by Mathematica
Time used: 7.587 (sec). Leaf size: 37
DSolve[x*(Log[x]-Log[y[x]])*y'[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -e^{c_1} W\left (-e^{-1-c_1} x\right ) \\ y(x)\to 0 \\ y(x)\to \frac {x}{e} \\ \end{align*}