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 : Monday, January 27, 2025 at 02:54:56 AM
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*}

✓ Solution by Maple

Time used: 0.220 (sec). Leaf size: 20

dsolve([diff(x(t),t$2)+2*diff(x(t),t)+2*x(t)=2*Dirac(t-Pi),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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