46.8.2 problem 4

Internal problem ID [7373]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.4, page 230
Problem number : 4
Date solved : Monday, January 27, 2025 at 02:51:13 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=4 \delta \left (t -3 \pi \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.382 (sec). Leaf size: 22

dsolve([diff(y(t),t$2)+16*y(t)=4*Dirac(t-3*Pi),y(0) = 2, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \operatorname {Heaviside}\left (t -3 \pi \right ) \sin \left (4 t \right )+2 \cos \left (4 t \right ) \]

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 23

DSolve[{D[y[t],{t,2}]+16*y[t]==4*DiracDelta[t-3*Pi],{y[0]==2,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \theta (t-3 \pi ) \sin (4 t)+2 \cos (4 t) \]