Internal problem ID [9306]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1727.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {2 y^{\prime \prime } y+\left (y^{\prime }\right )^{2}+1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.085 (sec). Leaf size: 823
dsolve(2*diff(diff(y(x),x),x)*y(x)+diff(y(x),x)^2+1=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} c_{2} \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2}-4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 x c_{2}+4 x^{2}\right )\right ) \left (-\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} c_{2} \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2}-4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 x c_{2}+4 x^{2}\right ) c_{1}+2 c_{2}+2 x \right )}{2}+\frac {c_{1}}{2} \\ y \relax (x ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} c_{2} \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2}-4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 x c_{2}+4 x^{2}\right )\right ) \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} c_{2} \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2}-4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 x c_{2}+4 x^{2}\right ) c_{1}-2 c_{2}-2 x \right )}{2}+\frac {c_{1}}{2} \\ y \relax (x ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} c_{2} \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+c_{1}^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} c_{2}+4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 x c_{2}+4 x^{2}\right )\right ) \left (-\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} c_{2} \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+c_{1}^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} c_{2}+4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 x c_{2}+4 x^{2}\right ) c_{1}-2 c_{2}-2 x \right )}{2}+\frac {c_{1}}{2} \\ y \relax (x ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} c_{2} \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+c_{1}^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} c_{2}+4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 x c_{2}+4 x^{2}\right )\right ) \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} c_{2} \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+c_{1}^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} c_{2}+4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 x c_{2}+4 x^{2}\right ) c_{1}+2 c_{2}+2 x \right )}{2}+\frac {c_{1}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.24 (sec). Leaf size: 129
DSolve[1 + y'[x]^2 + 2*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [e^{2 c_1} \text {ArcTan}\left (\frac {\sqrt {\text {$\#$1}}}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\right )-\sqrt {\text {$\#$1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\sqrt {\text {$\#$1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}-e^{2 c_1} \text {ArcTan}\left (\frac {\sqrt {\text {$\#$1}}}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\right )\&\right ][x+c_2] \\ \end{align*}