22.1.1 problem 1

Internal problem ID [4087]
Book : Applied Differential equations, Newby Curle. Van Nostrand Reinhold. 1972
Section : Examples, page 35
Problem number : 1
Date solved : Monday, January 27, 2025 at 08:08:17 AM
CAS classification : [_quadrature]

\begin{align*} y&=y^{\prime }+\frac {{y^{\prime }}^{2}}{2} \end{align*}

✓ Solution by Maple

Time used: 0.027 (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}^{x -1-c_{1}}\right ) \left (\operatorname {LambertW}\left (\sqrt {2}\, {\mathrm e}^{x -1-c_{1}}\right )+2\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 15.127 (sec). Leaf size: 66

DSolve[y[x]==D[y[x],x]+1/2*(D[y[x],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*}