7.21.5 problem 5

Internal problem ID [568]
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 : 5
Date solved : Saturday, March 29, 2025 at 04:56:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+2 x^{\prime }+2 x&=2 \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: 0.124 (sec). Leaf size: 20

ode:=diff(diff(x(t),t),t)+2*diff(x(t),t)+2*x(t) = 2*Dirac(t-Pi); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 22

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

\[ x(t)\to -2 e^{\pi -t} \theta (t-\pi ) \sin (t) \]

✓ Sympy. Time used: 2.427 (sec). Leaf size: 73

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-2*Dirac(t - pi) + 2*x(t) + 2*Derivative(x(t), 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)
 

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