Internal problem ID [860]
Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima,
Meade
Section: Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y-\left (\delta \left (-2 \pi +t \right )\right ) \cos \left (t \right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 15
dsolve([diff(y(t),t$2)+y(t)=Dirac(t-2*Pi)*cos(t),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
\[ y \left (t \right ) = \sin \left (t \right ) \left (\operatorname {Heaviside}\left (-2 \pi +t \right )+1\right ) \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 16
DSolve[{y''[t]+y[t]==DiracDelta[t-2*Pi]*Cos[t],{y[0]==0,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to (\theta (t-2 \pi )+1) \sin (t) \\ \end{align*}