63.17.3 problem 4

Internal problem ID [13202]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.4 Impulsive sources. Exercises page 173
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 05:12:20 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x&=\delta \left (t -2\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: 8.943 (sec). Leaf size: 13

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

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

✓ Solution by Mathematica

Time used: 0.059 (sec). Leaf size: 82

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

\[ x(t)\to -\sin (t) \int _1^0\cos (2) \delta (K[2]-2)dK[2]+\sin (t) \int _1^t\cos (2) \delta (K[2]-2)dK[2]-\cos (t) \int _1^0-\delta (K[1]-2) \sin (2)dK[1]+\cos (t) \int _1^t-\delta (K[1]-2) \sin (2)dK[1] \]