63.17.4 problem 6

Internal problem ID [13203]
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 : 6
Date solved : Tuesday, January 28, 2025 at 05:12:21 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x&=\delta \left (t -2\right )-\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: 10.032 (sec). Leaf size: 29

dsolve([diff(x(t),t$2)+4*x(t)=Dirac(t-2)-Dirac(t-5),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -5\right ) \sin \left (-10+2 t \right )}{2}+\frac {\operatorname {Heaviside}\left (t -2\right ) \sin \left (2 t -4\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.127 (sec). Leaf size: 140

DSolve[{D[x[t],{t,2}]+4*x[t]==DiracDelta[t-2]-DiracDelta[t-5],{x[0]==0,Derivative[1][x][0 ]==0}},x[t],t,IncludeSingularSolutions -> True]
 

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