Internal problem ID [2380]
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 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-4 y-\left (\delta \left (t -3\right )\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 38
dsolve([diff(y(t),t$2)-4*y(t)=Dirac(t-3),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
\[ y \relax (t ) = \frac {\left (-{\mathrm e}^{-2 t +6}+{\mathrm e}^{2 t -6}\right ) \theta \left (t -3\right )}{4}-\frac {{\mathrm e}^{-2 t}}{4}+\frac {{\mathrm e}^{2 t}}{4} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 26
DSolve[{y''[t]-4*y[t]==DiracDelta[t-3],{y[0]==0,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{2} (\sinh (2 t)-\theta (t-3) \sinh (6-2 t)) \\ \end{align*}