Internal problem ID [2998]
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]
\[ \boxed {y-y^{\prime }-\frac {{y^{\prime }}^{2}}{2}=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 102
dsolve(y(x)=diff(y(x),x)+1/2*(diff(y(x),x))^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {{\mathrm e}^{2 \operatorname {RootOf}\left (-\textit {\_Z} -2 x -2 \,{\mathrm e}^{\textit {\_Z}}-2+2 c_{1} -\ln \left (2\right )+\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}+2\right )^{2}\right )\right )}}{2}+{\mathrm e}^{\operatorname {RootOf}\left (-\textit {\_Z} -2 x -2 \,{\mathrm e}^{\textit {\_Z}}-2+2 c_{1} -\ln \left (2\right )+\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}+2\right )^{2}\right )\right )} \\ y \left (x \right ) &= \frac {\operatorname {LambertW}\left (\sqrt {2}\, {\mathrm e}^{-c_{1} +x -1}\right ) \left (\operatorname {LambertW}\left (\sqrt {2}\, {\mathrm e}^{-c_{1} +x -1}\right )+2\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 18.04 (sec). Leaf size: 66
DSolve[y[x]==y'[x]+1/2*(y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} W\left (-e^{x-1-c_1}\right ) \left (2+W\left (-e^{x-1-c_1}\right )\right ) \\ y(x)\to \frac {1}{2} W\left (e^{x-1+c_1}\right ) \left (2+W\left (e^{x-1+c_1}\right )\right ) \\ y(x)\to 0 \\ \end{align*}