Internal problem ID [5947]
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: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y-\left (\delta \left (t -\frac {\pi }{2}\right )\right )-\left (\delta \left (t -\frac {3 \pi }{2}\right )\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 22
dsolve([diff(y(t),t$2)+y(t)=Dirac(t-1/2*Pi)+Dirac(t-3/2*Pi),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \left (\operatorname {Heaviside}\left (t -\frac {3 \pi }{2}\right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \cos \left (t \right ) \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 27
DSolve[{y''[t]+y[t]==DiracDelta[t-1/2*Pi]+DiracDelta[t-3/2*Pi],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to (\theta (2 t-3 \pi )-\theta (2 t-\pi )) \cos (t) \\ \end{align*}