46.8.3 problem 5

Internal problem ID [7374]
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 : 5
Date solved : Monday, January 27, 2025 at 02:51:14 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\delta \left (t -\pi \right )-\delta \left (t -2 \pi \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.293 (sec). Leaf size: 25

dsolve([diff(y(t),t$2)+y(t)=Dirac(t-Pi)-Dirac(t-2*Pi),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = \sin \left (t \right ) \left (1-\operatorname {Heaviside}\left (t -2 \pi \right )-\operatorname {Heaviside}\left (t -\pi \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 23

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

\[ y(t)\to -((\theta (t-2 \pi )+\theta (t-\pi )-1) \sin (t)) \]