63.17.5 problem 7

Internal problem ID [13204]
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 : 7
Date solved : Tuesday, January 28, 2025 at 05:12:23 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x&=3 \delta \left (t -2 \pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 8.591 (sec). Leaf size: 17

dsolve([diff(x(t),t$2)+x(t)=3*Dirac(t-2*Pi),x(0) = 0, D(x)(0) = 1],x(t), singsol=all)
 

\[ x \left (t \right ) = \sin \left (t \right ) \left (3 \operatorname {Heaviside}\left (t -2 \pi \right )+1\right ) \]

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 47

DSolve[{D[x[t],{t,2}]+x[t]==3*DiracDelta[t-2*Pi],{x[0]==0,Derivative[1][x][0 ]==1}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \sin (t) \left (-\int _1^03 \delta (K[1]-2 \pi )dK[1]\right )+\sin (t) \int _1^t3 \delta (K[1]-2 \pi )dK[1]+\sin (t) \]