Internal problem ID [9302]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1723.
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 {y^{\prime \prime } \left (y+x \right )+\left (y^{\prime }\right )^{2}-y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.115 (sec). Leaf size: 16
dsolve(diff(diff(y(x),x),x)*(x+y(x))+diff(y(x),x)^2-diff(y(x),x)=0,y(x), singsol=all)
\[ y \relax (x ) = \sqrt {c_{1}+2 x}\, c_{2}+c_{1}+x \]
✓ Solution by Mathematica
Time used: 0.484 (sec). Leaf size: 110
DSolve[-y'[x] + y'[x]^2 + (x + y[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x+\frac {1}{2} e^{-2 c_1} \left (e^{c_1}-\sqrt {2} \sqrt {e^{2 c_1} \left (1+4 e^{c_1} (x+c_2)\right )}\right )+2 c_2 \\ y(x)\to x+\frac {1}{2} e^{-2 c_1} \left (\sqrt {2} \sqrt {e^{2 c_1} \left (1+4 e^{c_1} (x+c_2)\right )}+e^{c_1}\right )+2 c_2 \\ \end{align*}