Internal problem ID [6652]
Book: Second order enumerated odes
Section: section 1
Problem number: 19.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
Solve \begin {gather*} \boxed {\left (y^{\prime \prime }\right )^{2}+y^{\prime }-x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 122
dsolve(diff(y(x),x$2)^2+diff(y(x),x)=x,y(x), singsol=all)
\begin{align*} y \relax (x ) = \int \left (-{\mathrm e}^{2 \RootOf \left (\textit {\_Z} -x -2 \,{\mathrm e}^{\textit {\_Z}}+2+c_{1}-\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}-2\right )^{2}\right )\right )}+2 \,{\mathrm e}^{\RootOf \left (\textit {\_Z} -x -2 \,{\mathrm e}^{\textit {\_Z}}+2+c_{1}-\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}-2\right )^{2}\right )\right )}+x \right )d x -x +c_{2} \\ y \relax (x ) = \frac {2 \LambertW \left (-c_{1} {\mathrm e}^{-\frac {x}{2}-1}\right )^{3}}{3}+3 \LambertW \left (-c_{1} {\mathrm e}^{-\frac {x}{2}-1}\right )^{2}+4 \LambertW \left (-c_{1} {\mathrm e}^{-\frac {x}{2}-1}\right )+\frac {x^{2}}{2}-x +c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.273 (sec). Leaf size: 151
DSolve[(y''[x])^2+y'[x]==x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2}{3} \text {ProductLog}\left (e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right ){}^3+3 \text {ProductLog}\left (e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right ){}^2+4 \text {ProductLog}\left (e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right )+\frac {1}{2} (x-2) x+c_2 \\ y(x)\to \frac {2}{3} \text {ProductLog}\left (-e^{\frac {1}{2} (-x-2+c_1)}\right ){}^3+3 \text {ProductLog}\left (-e^{\frac {1}{2} (-x-2+c_1)}\right ){}^2+4 \text {ProductLog}\left (-e^{\frac {1}{2} (-x-2+c_1)}\right )+\frac {1}{2} (x-2) x+c_2 \\ \end{align*}