Internal problem ID [8198]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 618.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (1+y\right ) \left (\left (y-\ln \left (1+y\right )-\ln \relax (x )\right ) x +1\right )}{y x}=0} \end {gather*}
✓ Solution by Maple
Time used: 2.17 (sec). Leaf size: 62
dsolve(diff(y(x),x) = (y(x)+1)*((y(x)-ln(y(x)+1)-ln(x))*x+1)/y(x)/x,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {{\mathrm e}^{-\LambertW \left (-\frac {{\mathrm e}^{-1}}{x}\right )-1}-x}{x} \\ y \relax (x ) = \frac {{\mathrm e}^{-\LambertW \left (-\frac {{\mathrm e}^{-c_{1} {\mathrm e}^{x}-1}}{x}\right )-c_{1} {\mathrm e}^{x}-1}-x}{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.063 (sec). Leaf size: 25
DSolve[y'[x] == ((1 + y[x])*(1 + x*(-Log[x] - Log[1 + y[x]] + y[x])))/(x*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -1-\text {ProductLog}\left (-\frac {e^{-1+c_1 e^x}}{x}\right ) \\ \end{align*}