81.15.18 problem 19-19

Internal problem ID [21726]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 19. Change of variables. Page 483
Problem number : 19-19
Date solved : Thursday, October 02, 2025 at 08:01:25 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} y+\left (1+y^{2} {\mathrm e}^{2 x}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.099 (sec). Leaf size: 18

ode:=y(x)+(1+exp(2*x)*y(x)^2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-x}}{\sqrt {\operatorname {LambertW}\left (c_1 \,{\mathrm e}^{-2 x}\right )}} \]

✓ Mathematica. Time used: 1.995 (sec). Leaf size: 57

ode=y[x]+(1+y[x]^2*Exp[2*x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {e^{-x}}{\sqrt {W\left (e^{-2 x+2 c_1}\right )}}\\ y(x)&\to \frac {e^{-x}}{\sqrt {W\left (e^{-2 x+2 c_1}\right )}}\\ y(x)&\to 0 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((y(x)**2*exp(2*x) + 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : NoneType object is not subscriptable