Internal problem ID [9344]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1765.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {x y y^{\prime \prime }-2 x \left (y^{\prime }\right )^{2}+y^{\prime } y a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.055 (sec). Leaf size: 35
dsolve(x*y(x)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)^2+a*y(x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = -\frac {\left (a -1\right ) x^{a}}{c_{2} x^{a} a -c_{2} x^{a}-x c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.185 (sec). Leaf size: 24
DSolve[a*y[x]*y'[x] - 2*x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 x^a}{x+(a-1) c_1 x^a} \\ \end{align*}