Internal problem ID [2387]
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 13.
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-4 \sin \relax (t )-\left (\delta \left (t -\frac {\pi }{6}\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: 0.033 (sec). Leaf size: 64
dsolve([diff(y(t),t$2)+2*diff(y(t),t)+5*y(t)=4*sin(t)+Dirac(t-Pi/6),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
\[ y \relax (t ) = \frac {3 \,{\mathrm e}^{-t} \sin \left (2 t \right )}{10}+\frac {2 \,{\mathrm e}^{-t} \cos \left (2 t \right )}{5}-\frac {\theta \left (t -\frac {\pi }{6}\right ) \left (\sqrt {3}\, \cos \left (2 t \right )-\sin \left (2 t \right )\right ) {\mathrm e}^{-t +\frac {\pi }{6}}}{4}-\frac {2 \cos \relax (t )}{5}+\frac {4 \sin \relax (t )}{5} \]
✓ Solution by Mathematica
Time used: 0.216 (sec). Leaf size: 73
DSolve[{y''[t]+2*y'[t]+5*y[t]==4*Sin[t]+DiracDelta[t-Pi/6],{y[0]==0,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{20} e^{-t} \left (-5 e^{\pi /6} \theta (6 t-\pi ) \left (\sqrt {3} \cos (2 t)-\sin (2 t)\right )+6 \sin (2 t)+8 \cos (2 t)-8 e^t (\cos (t)-2 \sin (t))\right ) \\ \end{align*}