Internal problem ID [868]
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: 19(b).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+2 y^{\prime }+2 y-\left (\delta \left (-\pi +t \right )\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 20
dsolve([diff(y(t),t$2)+2*diff(y(t),t)+2*y(t)=Dirac(t-Pi),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = -\sin \relax (t ) \theta \left (-\pi +t \right ) {\mathrm e}^{\pi -t} \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 22
DSolve[{y''[t]+2*y'[t]+2*y[t]==DiracDelta[t-Pi],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -e^{\pi -t} \theta (t-\pi ) \sin (t) \\ \end{align*}