Internal problem ID [2381]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.8.
page 710
Problem number: Problem 7.
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 }+5 y-\left (\delta \left (t -\frac {\pi }{2}\right )\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 2] \end {align*}
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 32
dsolve([diff(y(t),t$2)+2*diff(y(t),t)+5*y(t)=Dirac(t-Pi/2),y(0) = 0, D(y)(0) = 2],y(t), singsol=all)
\[ y \relax (t ) = -\frac {\sin \left (2 t \right ) \left (\theta \left (t -\frac {\pi }{2}\right ) {\mathrm e}^{-t +\frac {\pi }{2}}-2 \,{\mathrm e}^{-t}\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.047 (sec). Leaf size: 34
DSolve[{y''[t]+2*y'[t]+5*y[t]==DiracDelta[t-Pi/2],{y[0]==0,y'[0]==2}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -e^{-t} \left (e^{\pi /2} \theta (2 t-\pi )-2\right ) \sin (t) \cos (t) \\ \end{align*}