16.1.1 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 : Tuesday, September 30, 2025 at 07:02:28 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.071 (sec). Leaf size: 106

ode:=y(x) = diff(y(x),x)+1/2*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\operatorname {LambertW}\left (-\sqrt {2}\, {\mathrm e}^{-1+x -c_1}\right ) \left (\operatorname {LambertW}\left (-\sqrt {2}\, {\mathrm e}^{-1+x -c_1}\right )+2\right )}{2} \\ y &= \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 )} \\ \end{align*}

✓ Mathematica. Time used: 11.291 (sec). Leaf size: 66

ode=y[x]==D[y[x],x]+1/2*(D[y[x],x])^2; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 0.540 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - Derivative(y(x), x)**2/2 - Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ x + \sqrt {2 y{\left (x \right )} + 1} - \log {\left (\sqrt {2 y{\left (x \right )} + 1} + 1 \right )} = C_{1}, \ x - \sqrt {2 y{\left (x \right )} + 1} - \log {\left (\sqrt {2 y{\left (x \right )} + 1} - 1 \right )} = C_{1}\right ] \]