Internal problem ID [9350]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1771.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {x^{2} \left (y+x \right ) y^{\prime \prime }-\left (y^{\prime } x -y\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.052 (sec). Leaf size: 27
dsolve(x^2*(x+y(x))*diff(diff(y(x),x),x)-(x*diff(y(x),x)-y(x))^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -x \\ y \relax (x ) = \frac {x \,{\mathrm e}^{\frac {c_{1}}{x}} {\mathrm e}^{-1}}{c_{2}}-x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.058 (sec). Leaf size: 20
DSolve[-(-y[x] + x*y'[x])^2 + x^2*(x + y[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x \left (-1+c_2 e^{\frac {c_1}{x}}\right ) \\ \end{align*}