Internal problem ID [2382]
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 8.
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^{\prime }+13 y-\left (\delta \left (t -\frac {\pi }{4}\right )\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 3, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.398 (sec). Leaf size: 53
dsolve([diff(y(t),t$2)-4*diff(y(t),t)+13*y(t)=Dirac(t-Pi/4),y(0) = 3, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = -\frac {\sqrt {2}\, \theta \left (t -\frac {\pi }{4}\right ) {\mathrm e}^{2 t -\frac {\pi }{2}} \left (\sin \left (3 t \right )+\cos \left (3 t \right )\right )}{6}+3 \,{\mathrm e}^{2 t} \left (\cos \left (3 t \right )-\frac {2 \sin \left (3 t \right )}{3}\right ) \]
✓ Solution by Mathematica
Time used: 0.103 (sec). Leaf size: 61
DSolve[{y''[t]-4*y'[t]+13*y[t]==DiracDelta[t-Pi/4],{y[0]==3,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{6} e^{2 t} \left (6 (3 \cos (3 t)-2 \sin (3 t))-\sqrt {2} e^{-\pi /2} \theta (4 t-\pi ) (\sin (3 t)+\cos (3 t))\right ) \\ \end{align*}