Internal problem ID [10067]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1745 (book 6.154).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {2 y^{\prime \prime } y-{y^{\prime }}^{2} \left ({y^{\prime }}^{2}+1\right )=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 331
dsolve(2*diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2*(diff(y(x),x)^2+1)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (\operatorname {RootOf}\left (\left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right ) \left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right )\right ) c_{1} -2 x -2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right ) \left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right )\right )\right )}{2}+\frac {c_{1}}{2} \\ y \left (x \right ) &= \frac {\left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right )\right ) c_{1} +2 x +2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right )\right )\right )}{2}+\frac {c_{1}}{2} \\ y \left (x \right ) &= \frac {\left (-\operatorname {RootOf}\left (\left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right ) \left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right )\right ) c_{1} +2 x +2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right ) \left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right )\right )\right )}{2}+\frac {c_{1}}{2} \\ y \left (x \right ) &= \frac {\left (-\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right )\right ) c_{1} -2 x -2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right )\right )\right )}{2}+\frac {c_{1}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.67 (sec). Leaf size: 501
DSolve[-(y'[x]^2*(1 + y'[x]^2)) + 2*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-i e^{-c_1} \left (\sqrt {\text {$\#$1}} \sqrt {-1+\text {$\#$1} e^{2 c_1}}-e^{-c_1} \text {arctanh}\left (\frac {e^{-c_1} \sqrt {-1+\text {$\#$1} e^{2 c_1}}}{\sqrt {\text {$\#$1}}}\right )\right )\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [i e^{-c_1} \left (\sqrt {\text {$\#$1}} \sqrt {-1+\text {$\#$1} e^{2 c_1}}-e^{-c_1} \text {arctanh}\left (\frac {e^{-c_1} \sqrt {-1+\text {$\#$1} e^{2 c_1}}}{\sqrt {\text {$\#$1}}}\right )\right )\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-i e^{-(-c_1)} \left (\sqrt {\text {$\#$1}} \sqrt {-1+\text {$\#$1} e^{2 (-c_1)}}-e^{-(-c_1)} \text {arctanh}\left (\frac {e^{-(-c_1)} \sqrt {-1+\text {$\#$1} e^{2 (-c_1)}}}{\sqrt {\text {$\#$1}}}\right )\right )\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [i e^{-(-c_1)} \left (\sqrt {\text {$\#$1}} \sqrt {-1+\text {$\#$1} e^{2 (-c_1)}}-e^{-(-c_1)} \text {arctanh}\left (\frac {e^{-(-c_1)} \sqrt {-1+\text {$\#$1} e^{2 (-c_1)}}}{\sqrt {\text {$\#$1}}}\right )\right )\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-i e^{-c_1} \left (\sqrt {\text {$\#$1}} \sqrt {-1+\text {$\#$1} e^{2 c_1}}-e^{-c_1} \text {arctanh}\left (\frac {e^{-c_1} \sqrt {-1+\text {$\#$1} e^{2 c_1}}}{\sqrt {\text {$\#$1}}}\right )\right )\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [i e^{-c_1} \left (\sqrt {\text {$\#$1}} \sqrt {-1+\text {$\#$1} e^{2 c_1}}-e^{-c_1} \text {arctanh}\left (\frac {e^{-c_1} \sqrt {-1+\text {$\#$1} e^{2 c_1}}}{\sqrt {\text {$\#$1}}}\right )\right )\&\right ][x+c_2] \\ \end{align*}