77.1.53 problem 72 (page 112)
Internal
problem
ID
[17864]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
72
(page
112)
Date
solved
:
Thursday, March 13, 2025 at 11:04:41 AM
CAS
classification
:
[_quadrature]
\begin{align*} y&={\mathrm e}^{y^{\prime }} {y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.319 (sec). Leaf size: 38
ode:=y(x) = exp(diff(y(x),x))*diff(y(x),x)^2;
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: 0.322 (sec). Leaf size: 102
ode=y[x]==Exp[D[y[x],x]]*D[y[x],x]^2;
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: 1.124 (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)
\[
\left [ - 2 x + \frac {y{\left (x \right )}}{W\left (- \frac {\sqrt {y{\left (x \right )}}}{2}\right )} + \frac {y{\left (x \right )}}{2 W^{2}\left (- \frac {\sqrt {y{\left (x \right )}}}{2}\right )} = C_{1}, \ - 2 x + \frac {y{\left (x \right )}}{W\left (\frac {\sqrt {y{\left (x \right )}}}{2}\right )} + \frac {y{\left (x \right )}}{2 W^{2}\left (\frac {\sqrt {y{\left (x \right )}}}{2}\right )} = C_{1}\right ]
\]