46.8.1 problem 3

Internal problem ID [7372]
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 : 3
Date solved : Monday, January 27, 2025 at 02:51:13 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=8\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.269 (sec). Leaf size: 23

dsolve([diff(y(t),t$2)+4*y(t)=Dirac(t-Pi),y(0) = 8, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\operatorname {Heaviside}\left (t -\pi \right ) \sin \left (2 t \right )}{2}+8 \cos \left (2 t \right ) \]

✓ Solution by Mathematica

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

DSolve[{D[y[t],{t,2}]+4*y[t]==DiracDelta[t-Pi],{y[0]==8,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \theta (t-\pi ) \sin (t) \cos (t)+8 \cos (2 t) \]