Internal problem ID [2489]
Book: Applied Differential equations, N Curle, 1971
Section: Examples, page 35
Problem number: 1.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {y-y^{\prime }-\frac {\left (y^{\prime }\right )^{2}}{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 139
dsolve(y(x)=diff(y(x),x)+1/2*(diff(y(x),x))^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {{\mathrm e}^{2 \RootOf \left (-\textit {\_Z} -2 x -2 \,{\mathrm e}^{\textit {\_Z}}-2+2 c_{1}+\ln \left (\frac {{\mathrm e}^{3 \textit {\_Z}}}{2}+2 \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}\right )\right )}}{2}+{\mathrm e}^{\RootOf \left (-\textit {\_Z} -2 x -2 \,{\mathrm e}^{\textit {\_Z}}-2+2 c_{1}+\ln \left (\frac {{\mathrm e}^{3 \textit {\_Z}}}{2}+2 \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}\right )\right )} \\ y \relax (x ) = \frac {{\mathrm e}^{-2 \LambertW \left (\sqrt {2}\, {\mathrm e}^{-c_{1}+x -1}\right )-2 c_{1}+2 x +\ln \relax (2)-2}}{2}+{\mathrm e}^{-\LambertW \left ({\mathrm e}^{-c_{1}} {\mathrm e}^{x} \sqrt {2}\, {\mathrm e}^{-1}\right )-c_{1}+x +\frac {\ln \relax (2)}{2}-1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.067 (sec). Leaf size: 61
DSolve[y[x]==y'[x]+1/2*(y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \text {ProductLog}\left (-e^{x-1-c_1}\right ) \left (2+\text {ProductLog}\left (-e^{x-1-c_1}\right )\right ) \\ y(x)\to \frac {1}{2} \text {ProductLog}\left (e^{x-1+c_1}\right ) \left (2+\text {ProductLog}\left (e^{x-1+c_1}\right )\right ) \\ \end{align*}