Internal problem ID [6679]
Book: Second order enumerated odes
Section: section 1
Problem number: 46.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y \left (y^{\prime \prime }\right )^{2}+\left (y^{\prime }\right )^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.516 (sec). Leaf size: 166
dsolve(y(x)*diff(y(x),x$2)^2+diff(y(x),x)^3=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = c_{1} \\ y \relax (x ) = 0 \\ y \relax (x ) = \frac {c_{2} \left (\LambertW \left (c_{1} {\mathrm e}^{-1+\frac {x}{2}}\right )+1\right )^{2}}{\LambertW \left (c_{1} {\mathrm e}^{-1+\frac {x}{2}}\right )^{2}} \\ y \relax (x ) = \frac {c_{2} \left (\LambertW \left (-c_{1} {\mathrm e}^{-1+\frac {x}{2}}\right )+1\right )^{2}}{\LambertW \left (-c_{1} {\mathrm e}^{-1+\frac {x}{2}}\right )^{2}} \\ y \relax (x ) = {\mathrm e}^{-\left (\int {\mathrm e}^{2 \RootOf \left ({\mathrm e}^{\textit {\_Z}} \ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} {\mathrm e}^{\textit {\_Z}}-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} x +\ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1}-2 \textit {\_Z} +x -2\right )}d x \right )-2 \left (\int {\mathrm e}^{\RootOf \left ({\mathrm e}^{\textit {\_Z}} \ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} {\mathrm e}^{\textit {\_Z}}-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} x +\ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1}-2 \textit {\_Z} +x -2\right )}d x \right )-x +c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.916 (sec). Leaf size: 117
DSolve[y[x]*y''[x]^2+y'[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-4 \left (\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}-i c_1\right )-\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}-i c_1\right )}\right )\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-4 \left (\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}+i c_1\right )}+\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}+i c_1\right )\right )\&\right ][x+c_2] \\ \end{align*}