Internal problem ID [5949]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y-\left (\delta \left (-2 \pi +t \right )\right )-\left (\delta \left (t -4 \pi \right )\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 1, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 23
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)
\[ y \relax (t ) = \sin \relax (t ) \theta \left (-2 \pi +t \right )+\sin \relax (t ) \theta \left (t -4 \pi \right )+\cos \relax (t ) \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 24
DSolve[{y''[t]+y[t]==DiracDelta[t-2*Pi]+DiracDelta[t-4*Pi],{y[0]==1,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to (\theta (t-4 \pi )+\theta (t-2 \pi )) \sin (t)+\cos (t) \\ \end{align*}