Internal problem ID [6651]
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]]
\[ \boxed {{y^{\prime \prime }}^{2}+y^{\prime }-x=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 122
dsolve(diff(y(x),x$2)^2+diff(y(x),x)=x,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \int \left (-{\mathrm e}^{2 \operatorname {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}^{\operatorname {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 \left (x \right ) = \frac {2 \operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-\frac {x}{2}-1}\right )^{3}}{3}+3 \operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-\frac {x}{2}-1}\right )^{2}+4 \operatorname {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: 23.232 (sec). Leaf size: 166
DSolve[(y''[x])^2+y'[x]==x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2}{3} W\left (e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right ){}^3+3 W\left (e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right ){}^2+4 W\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} W\left (-e^{\frac {1}{2} (-x-2+c_1)}\right ){}^3+3 W\left (-e^{\frac {1}{2} (-x-2+c_1)}\right ){}^2+4 W\left (-e^{\frac {1}{2} (-x-2+c_1)}\right )+\frac {1}{2} (x-2) x+c_2 \\ y(x)\to \frac {1}{2} (x-2) x+c_2 \\ \end{align*}