90.8.4 problem 4

Internal problem ID [25178]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 2. The Laplace Transform. Exercises at page 109
Problem number : 4
Date solved : Thursday, October 02, 2025 at 11:57:36 PM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 13

ode:=diff(y(t),t)+a*y(t) = exp(-a*t); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 15

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

\begin{align*} y(t)&\to (t+1) e^{-a t} \end{align*}

✓ Sympy. Time used: 0.089 (sec). Leaf size: 10

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

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