75.8.10 problem 208

Internal problem ID [16755]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8. First order not solved for the derivative. Exercises page 67
Problem number : 208
Date solved : Monday, March 31, 2025 at 03:16:04 PM
CAS classification : [_quadrature]

\begin{align*} y&={y^{\prime }}^{2} {\mathrm e}^{y^{\prime }} \end{align*}

✓ Maple. Time used: 0.435 (sec). Leaf size: 38

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

\begin{align*} y &= 0 \\ y &= \frac {\left (x -c_1 \right ) \left (\operatorname {LambertW}\left (\left (x -c_1 \right ) {\mathrm e}\right )-1\right )^{2}}{\operatorname {LambertW}\left (\left (x -c_1 \right ) {\mathrm e}\right )} \\ \end{align*}

✓ Mathematica. Time used: 0.313 (sec). Leaf size: 102

ode=y[x]==D[y[x],x]^2*Exp[D[y[x],x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1}}{W\left (-\frac {\sqrt {\text {$\#$1}}}{2}\right )}+\frac {\text {$\#$1}}{2 W\left (-\frac {\sqrt {\text {$\#$1}}}{2}\right )^2}\&\right ][2 x+c_1] \\ y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1}}{W\left (\frac {\sqrt {\text {$\#$1}}}{2}\right )}+\frac {\text {$\#$1}}{2 W\left (\frac {\sqrt {\text {$\#$1}}}{2}\right )^2}\&\right ][2 x+c_1] \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.119 (sec). Leaf size: 71

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

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