80.10.7 problem 30

Internal problem ID [21402]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 11. Laplace transform. Excercise 11.7 at page 248
Problem number : 30
Date solved : Thursday, October 02, 2025 at 07:30:50 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} x^{\prime }+x&=\operatorname {Heaviside}\left (t -a \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.092 (sec). Leaf size: 41

ode:=diff(x(t),t)+x(t) = Heaviside(t-a); 
ic:=[x(0) = 0]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 39

ode=D[x[t],t]+x[t]==UnitStep[t-a]; 
ic={x[0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-t} \left (\left (e^a-1\right ) \theta (-a)-\left (e^a-e^t\right ) \theta (t-a)\right ) \end{align*}

✓ Sympy. Time used: 0.241 (sec). Leaf size: 41

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

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