Internal problem ID [9396]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1817.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (y^{\prime } x -y\right ) y^{\prime \prime }+4 \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.064 (sec). Leaf size: 44
dsolve((x*diff(y(x),x)-y(x))*diff(diff(y(x),x),x)+4*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = {\mathrm e}^{\int _{}^{\ln \relax (x )}\left ({\mathrm e}^{\RootOf \left (\ln \left ({\mathrm e}^{\textit {\_Z}}-1\right ) {\mathrm e}^{\textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-\textit {\_b} \,{\mathrm e}^{\textit {\_Z}}+2\right )}-1\right )d \textit {\_b} +c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 9.143 (sec). Leaf size: 40
DSolve[4*y'[x]^2 + (-y[x] + x*y'[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 c_2 \text {ProductLog}\left (\frac {2 x}{e^2 c_1}\right ) \left (2+\text {ProductLog}\left (\frac {2 x}{e^2 c_1}\right )\right )}{4 x} \\ \end{align*}