Internal problem ID [9347]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1768.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {x \left (y+x \right ) y^{\prime \prime }+x \left (y^{\prime }\right )^{2}+\left (x -y\right ) y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 49
dsolve(x*(x+y(x))*diff(diff(y(x),x),x)+x*diff(y(x),x)^2+(x-y(x))*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -x \\ y \relax (x ) = -x -\sqrt {-c_{2} x^{2}+x^{2}+c_{1}} \\ y \relax (x ) = -x +\sqrt {-c_{2} x^{2}+x^{2}+c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 53
DSolve[-y[x] + (x - y[x])*y'[x] + x*y'[x]^2 + x*(x + y[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x-\sqrt {(1+2 c_2) x^2+c_1} \\ y(x)\to -x+\sqrt {(1+2 c_2) x^2+c_1} \\ \end{align*}