Internal problem ID [9304]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1725.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime } \left (x -y\right )-\left (y^{\prime }+1\right ) \left (\left (y^{\prime }\right )^{2}+1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.088 (sec). Leaf size: 106
dsolve(diff(diff(y(x),x),x)*(x-y(x))-(diff(y(x),x)+1)*(diff(y(x),x)^2+1)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = x +\RootOf \left (-x +\int _{}^{\textit {\_Z}}-\frac {c_{1}^{2} \textit {\_f}^{2}-1}{c_{1}^{2} \textit {\_f}^{2}+\sqrt {-c_{1}^{2} \textit {\_f}^{2}+2}\, c_{1} \textit {\_f} -2}d \textit {\_f} +c_{2}\right ) \\ y \relax (x ) = x +\RootOf \left (-x +\int _{}^{\textit {\_Z}}-\frac {c_{1}^{2} \textit {\_f}^{2}-1}{c_{1}^{2} \textit {\_f}^{2}-\sqrt {-c_{1}^{2} \textit {\_f}^{2}+2}\, c_{1} \textit {\_f} -2}d \textit {\_f} +c_{2}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.774 (sec). Leaf size: 59
DSolve[(-1 - y'[x])*(1 + y'[x]^2) + (x - y[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {e^{2 c_1}-(x+c_2){}^2}-c_2 \\ y(x)\to \sqrt {e^{2 c_1}-(x+c_2){}^2}-c_2 \\ \end{align*}