Internal problem ID [857]
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: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y-\left (\delta \left (-\pi +t \right )\right )+\delta \left (-2 \pi +t \right )=0} \] 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: 25
dsolve([diff(y(t),t$2)+4*y(t)=Dirac(t-Pi)-Dirac(t-2*Pi),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = -\frac {\left (\operatorname {Heaviside}\left (-2 \pi +t \right )-\operatorname {Heaviside}\left (-\pi +t \right )\right ) \sin \left (2 t \right )}{2} \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 25
DSolve[{y''[t]+4*y[t]==DiracDelta[t-Pi]-DiracDelta[t-2*Pi],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to (\theta (t-\pi )-\theta (t-2 \pi )) \sin (t) \cos (t) \\ \end{align*}