Internal problem ID [9294]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1715.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y^{\prime \prime } y-a \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.055 (sec). Leaf size: 29
dsolve(diff(diff(y(x),x),x)*y(x)-a*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \left (\frac {1}{\left (-a +1\right ) \left (x c_{1}+c_{2}\right )}\right )^{\frac {1}{a -1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.096 (sec). Leaf size: 26
DSolve[-(a*y'[x]^2) + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 (-a x+x-c_1){}^{\frac {1}{1-a}} \\ \end{align*}