Internal problem ID [9338]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1760 (book 6.169).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _nonlinear], _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 }+x \left (y^{\prime }\right )^{2}-y^{\prime } y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 35
dsolve(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 ) = \sqrt {c_{1} x^{2}+2 c_{2}} \\ y \relax (x ) = -\sqrt {c_{1} x^{2}+2 c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.099 (sec). Leaf size: 18
DSolve[-(y[x]*y'[x]) + x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \sqrt {x^2+c_1} \\ \end{align*}