57.5.27 problem 15(b)

Internal problem ID [14382]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.4.1. Integrating factors. Exercises page 41
Problem number : 15(b)
Date solved : Thursday, October 02, 2025 at 09:35:54 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Bernoulli]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

ode:=diff(x(t),t) = x(t)*(1+x(t)*exp(t)); 
dsolve(ode,x(t), singsol=all);
 

\[ x = -\frac {2 \,{\mathrm e}^{t}}{{\mathrm e}^{2 t}-2 c_1} \]

✓ Mathematica. Time used: 0.124 (sec). Leaf size: 27

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

\begin{align*} x(t)&\to -\frac {2 e^t}{e^{2 t}-2 c_1}\\ x(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.143 (sec). Leaf size: 14

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

\[ x{\left (t \right )} = \frac {2 e^{t}}{C_{1} - e^{2 t}} \]