Internal problem ID [9302]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1728 (book 6.137).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {2 y^{\prime \prime } y+{y^{\prime }}^{2}+1=0} \]
✓ Solution by Maple
Time used: 0.218 (sec). Leaf size: 1579
dsolve(2*diff(diff(y(x),x),x)*y(x)+diff(y(x),x)^2+1=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2} -4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right ) \left (-\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2} -4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right ) c_{1} +4 c_{2} +4 x \right )}{2}-\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2} -4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right ) c_{2} -\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2} -4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right ) x +\frac {c_{1}}{2} \\ y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} c_{2} +4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right ) \left (-\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} c_{2} +4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right ) c_{1} -4 c_{2} -4 x \right )}{2}+\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} c_{2} +4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right ) c_{2} +\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} c_{2} +4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right ) x +\frac {c_{1}}{2} \\ y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2} -4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right ) \operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2} -4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right ) c_{1}}{2}-\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2} -4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right ) c_{2} -\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2} -4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right ) x +\frac {c_{1}}{2} \\ y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} c_{2} +4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right ) \operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} c_{2} +4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right ) c_{1}}{2}+\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} c_{2} +4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right ) c_{2} +\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} c_{2} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} c_{2} +4 x c_{1} \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right ) x +\frac {c_{1}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.352 (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} \arctan \left (\frac {\sqrt {-\text {$\#$1}+e^{2 c_1}}}{\sqrt {\text {$\#$1}}}\right )-\sqrt {\text {$\#$1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [e^{2 c_1} \arctan \left (\frac {\sqrt {-\text {$\#$1}+e^{2 c_1}}}{\sqrt {\text {$\#$1}}}\right )+\sqrt {\text {$\#$1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}\&\right ][x+c_2] \\ \end{align*}