Internal problem ID [8145]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 565.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \relax (y)-y x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.375 (sec). Leaf size: 17
dsolve(y(x)*ln(diff(y(x),x))+diff(y(x),x)-y(x)*ln(y(x))-x*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\frac {\LambertW \left ({\mathrm e}^{x}\right ) \left (\LambertW \left ({\mathrm e}^{x}\right )+2\right )}{2}} \]
✓ Solution by Mathematica
Time used: 0.096 (sec). Leaf size: 24
DSolve[-(x*y[x]) - Log[y[x]]*y[x] + Log[y'[x]]*y[x] + y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^{\frac {1}{2} \text {ProductLog}\left (e^x\right ) \left (\text {ProductLog}\left (e^x\right )+2\right )} \\ \end{align*}