Internal problem ID [9348]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1770 (book 6.179).
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 {2 x y y^{\prime \prime }-x \left (y^{\prime }\right )^{2}+y^{\prime } y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 25
dsolve(2*x*y(x)*diff(diff(y(x),x),x)-x*diff(y(x),x)^2+y(x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = c_{1} \sqrt {x}\, c_{2}+c_{1}^{2} x +\frac {c_{2}^{2}}{4} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.109 (sec). Leaf size: 18
DSolve[y[x]*y'[x] - x*y'[x]^2 + 2*x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \left (\sqrt {x}+c_1\right ){}^2 \\ \end{align*}