Internal problem ID [9278]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1699.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y^{\prime \prime } y+\left (y^{\prime }\right )^{2}-y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.086 (sec). Leaf size: 37
dsolve(diff(diff(y(x),x),x)*y(x)+diff(y(x),x)^2-diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = -c_{1} \left (\LambertW \left (-\frac {{\mathrm e}^{-1} {\mathrm e}^{-\frac {c_{2}}{c_{1}}} {\mathrm e}^{-\frac {x}{c_{1}}}}{c_{1}}\right )+1\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.095 (sec). Leaf size: 32
DSolve[-y'[x] + y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -c_1 \left (1+\text {ProductLog}\left (-\frac {e^{-\frac {x+c_1+c_2}{c_1}}}{c_1}\right )\right ) \\ \end{align*}