7.21.6 problem 6

Internal problem ID [569]
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 : 6
Date solved : Monday, January 27, 2025 at 02:54:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+9 x&=\delta \left (t -3 \pi \right )+\cos \left (3 t \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: 0.319 (sec). Leaf size: 20

dsolve([diff(x(t),t$2)+9*x(t)=Dirac(t-3*Pi)+cos(3*t),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\sin \left (3 t \right ) \left (-2 \operatorname {Heaviside}\left (t -3 \pi \right )+t \right )}{6} \]

✓ Solution by Mathematica

Time used: 0.082 (sec). Leaf size: 23

DSolve[{D[x[t],{t,2}]+9*x[t]==DiracDelta[t-3*Pi]+Cos[3*t],{x[0]==0,Derivative[1][x][0] ==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {1}{6} (t-2 \theta (t-3 \pi )) \sin (3 t) \]