90.3.9 problem 9

Internal problem ID [25073]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 1. First order differential equations. Exercises at page 41
Problem number : 9
Date solved : Thursday, October 02, 2025 at 11:49:11 PM
CAS classification : [_separable]

\begin{align*} {\mathrm e}^{t} y^{\prime }&=y^{3}-y \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 35

ode:=exp(t)*diff(y(t),t) = y(t)^3-y(t); 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= \frac {1}{\sqrt {{\mathrm e}^{-2 \,{\mathrm e}^{-t}} c_1 +1}} \\ y &= -\frac {1}{\sqrt {{\mathrm e}^{-2 \,{\mathrm e}^{-t}} c_1 +1}} \\ \end{align*}

✓ Mathematica. Time used: 1.982 (sec). Leaf size: 132

ode=Exp[t]*D[y[t],{t,1}]==y[t]^3-y[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\frac {e^{e^{-t}}}{\sqrt {e^{2 e^{-t}}+e^{2 c_1}}}\\ y(t)&\to \frac {e^{e^{-t}}}{\sqrt {e^{2 e^{-t}}+e^{2 c_1}}}\\ y(t)&\to -1\\ y(t)&\to 0\\ y(t)&\to 1\\ y(t)&\to -\frac {e^{e^{-t}}}{\sqrt {e^{2 e^{-t}}}}\\ y(t)&\to \frac {e^{e^{-t}}}{\sqrt {e^{2 e^{-t}}}} \end{align*}

✓ Sympy. Time used: 1.680 (sec). Leaf size: 73

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t)**3 + y(t) + exp(t)*Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ \left [ y{\left (t \right )} = - \frac {\sqrt {- \frac {1}{C_{1} - e^{e^{- t}}}} e^{e^{- t}}}{\sqrt {C_{1} + e^{e^{- t}}}}, \ y{\left (t \right )} = \frac {\sqrt {- \frac {1}{C_{1} - e^{e^{- t}}}} e^{e^{- t}}}{\sqrt {C_{1} + e^{e^{- t}}}}\right ] \]