63.17.2 problem 3

Internal problem ID [13201]
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 : 3
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 -5\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: 8.812 (sec). Leaf size: 13

dsolve([diff(x(t),t$2)-x(t)=Dirac(t-5),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = \operatorname {Heaviside}\left (t -5\right ) \sinh \left (t -5\right ) \]

✓ Solution by Mathematica

Time used: 0.051 (sec). Leaf size: 104

DSolve[{D[x[t],{t,2}]-x[t]==DiracDelta[t-5],{x[0]==0,Derivative[1][x][0 ]==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to -e^{-t} \left (e^{2 t} \int _1^0\frac {\delta (K[1]-5)}{2 e^5}dK[1]-e^{2 t} \int _1^t\frac {\delta (K[1]-5)}{2 e^5}dK[1]-\int _1^t-\frac {1}{2} e^5 \delta (K[2]-5)dK[2]+\int _1^0-\frac {1}{2} e^5 \delta (K[2]-5)dK[2]\right ) \]