68.5.32 problem 32

Internal problem ID [17283]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.3, page 49
Problem number : 32
Date solved : Thursday, October 02, 2025 at 02:00:50 PM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 18

ode:=(1+exp(t))*diff(y(t),t)+exp(t)*y(t) = t; 
ic:=[y(0) = -1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {t^{2}-4}{2 \,{\mathrm e}^{t}+2} \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 21

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

\begin{align*} y(t)&\to \frac {t^2-4}{2 \left (e^t+1\right )} \end{align*}

✓ Sympy. Time used: 0.179 (sec). Leaf size: 14

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

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