Internal problem ID [856]
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: 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+2 y-\left (\delta \left (-\pi +t \right )\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 31
dsolve([diff(y(t),t$2)+2*diff(y(t),t)+2*y(t)=Dirac(t-Pi),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = -\sin \left (t \right ) \operatorname {Heaviside}\left (-\pi +t \right ) {\mathrm e}^{\pi -t}+{\mathrm e}^{-t} \left (\cos \left (t \right )+\sin \left (t \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.028 (sec). Leaf size: 29
DSolve[{y''[t]+2*y'[t]+2*y[t]==DiracDelta[t-Pi],{y[0]==1,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to e^{-t} \left (-e^{\pi } \theta (t-\pi ) \sin (t)+\sin (t)+\cos (t)\right ) \\ \end{align*}