63.15.13 problem 14

Internal problem ID [13125]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.2.1 Initial value problems. Exercises page 156
Problem number : 14
Date solved : Monday, March 31, 2025 at 07:34:54 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} x^{\prime }&=-x+\operatorname {Heaviside}\left (t -1\right )-\operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.139 (sec). Leaf size: 42

ode:=diff(x(t),t) = -x(t)+Heaviside(t-1)-Heaviside(t-2); 
ic:=x(0) = 1; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x = -\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{1-t}+\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{2-t}+\operatorname {Heaviside}\left (t -1\right )-\operatorname {Heaviside}\left (t -2\right )+{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 48

ode=D[x[t],t]==-x[t]+UnitStep[t-1]-UnitStep[t-2]; 
ic={x[0]==1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-t} & t\leq 1 \\ e^{-t} \left (1-e+e^2\right ) & t>2 \\ e^{-t} \left (1-e+e^t\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.436 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) + Heaviside(t - 2) - Heaviside(t - 1) + Derivative(x(t), t),0) 
ics = {x(0): 1} 
dsolve(ode,func=x(t),ics=ics)
 

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