7.21.2 problem 2

Internal problem ID [565]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.6 (Impulses and Delta functions). Problems at page 324
Problem number : 2
Date solved : Saturday, March 29, 2025 at 04:56:47 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x&=\delta \left (t \right )+\delta \left (t -\pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.211 (sec). Leaf size: 18

ode:=diff(diff(x(t),t),t)+4*x(t) = Dirac(t)+Dirac(t-Pi); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x = \frac {\sin \left (2 t \right ) \left (1+\operatorname {Heaviside}\left (t -\pi \right )\right )}{2} \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 26

ode=D[x[t],{t,2}]+4*x[t]==DiracDelta[t]+DiracDelta[t-Pi]; 
ic={x[0]==0,Derivative[1][x][0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to (-\theta (t)-\theta (t-\pi )+\theta (0)) \sin (t) (-\cos (t)) \]

✓ Sympy. Time used: 2.551 (sec). Leaf size: 97

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-Dirac(t) - Dirac(t - pi) + 4*x(t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (- \frac {\int \left (\operatorname {Dirac}{\left (t \right )} + \operatorname {Dirac}{\left (t - \pi \right )}\right ) \sin {\left (2 t \right )}\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t \right )} \sin {\left (2 t \right )}\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \pi \right )} \sin {\left (2 t \right )}\, dt}{2}\right ) \cos {\left (2 t \right )} + \left (\frac {\int \left (\operatorname {Dirac}{\left (t \right )} + \operatorname {Dirac}{\left (t - \pi \right )}\right ) \cos {\left (2 t \right )}\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t \right )} \cos {\left (2 t \right )}\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \pi \right )} \cos {\left (2 t \right )}\, dt}{2}\right ) \sin {\left (2 t \right )} \]