Internal problem ID [7435]
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]]
\[ \boxed {y {y^{\prime \prime }}^{2}+{y^{\prime }}^{3}=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 166
dsolve(y(x)*diff(y(x),x$2)^2+diff(y(x),x)^3=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = c_{1} y \left (x \right ) = 0 y \left (x \right ) = \frac {c_{2} {\left (\operatorname {LambertW}\left (c_{1} {\mathrm e}^{-1+\frac {x}{2}}\right )+1\right )}^{2}}{\operatorname {LambertW}\left (c_{1} {\mathrm e}^{-1+\frac {x}{2}}\right )^{2}} y \left (x \right ) = \frac {c_{2} {\left (\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-1+\frac {x}{2}}\right )+1\right )}^{2}}{\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-1+\frac {x}{2}}\right )^{2}} y \left (x \right ) = {\mathrm e}^{-\left (\int {\mathrm e}^{2 \operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}} \ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} {\mathrm e}^{\textit {\_Z}}-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+x \,{\mathrm e}^{\textit {\_Z}}+\ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} -2 \textit {\_Z} +x -2\right )}d x \right )-2 \left (\int {\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}} \ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} {\mathrm e}^{\textit {\_Z}}-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+x \,{\mathrm e}^{\textit {\_Z}}+\ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} -2 \textit {\_Z} +x -2\right )}d x \right )-x +c_{2}} \end{align*}
✓ Solution by Mathematica
Time used: 2.165 (sec). Leaf size: 361
DSolve[y[x]*y''[x]^2+y'[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-4 \left (\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}-i c_1\right )-\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}-i c_1\right )}\right )\&\right ][x+c_2] y(x)\to \text {InverseFunction}\left [-4 \left (\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}+i c_1\right )}+\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}+i c_1\right )\right )\&\right ][x+c_2] y(x)\to \text {InverseFunction}\left [-4 \left (\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}-i (-c_1)\right )-\frac {i (-c_1)}{2 \left (2 \sqrt {\text {$\#$1}}-i (-c_1)\right )}\right )\&\right ][x+c_2] y(x)\to \text {InverseFunction}\left [-4 \left (\frac {i (-c_1)}{2 \left (2 \sqrt {\text {$\#$1}}+i (-1) c_1\right )}+\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}+i (-1) c_1\right )\right )\&\right ][x+c_2] y(x)\to \text {InverseFunction}\left [-4 \left (\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}-i c_1\right )-\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}-i c_1\right )}\right )\&\right ][x+c_2] y(x)\to \text {InverseFunction}\left [-4 \left (\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}+i c_1\right )}+\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}+i c_1\right )\right )\&\right ][x+c_2] \end{align*}