52.8.6 problem 6

Internal problem ID [8370]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. EXERCISES 7.5. Page 315
Problem number : 6
Date solved : Monday, January 27, 2025 at 03:49:41 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\delta \left (t -2 \pi \right )+\delta \left (t -4 \pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.722 (sec). Leaf size: 25

dsolve([diff(y(t),t$2)+y(t)=Dirac(t-2*Pi)+Dirac(t-4*Pi),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 26

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

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