Internal problem ID [2896]
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]]
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=4 \sin \left (t \right )+\delta \left (t -\frac {\pi }{6}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 3.406 (sec). Leaf size: 56
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)
\[ y \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -\frac {\pi }{6}\right ) \left (\sqrt {3}\, \cos \left (t \right )^{2}-\cos \left (t \right ) \sin \left (t \right )-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{-t +\frac {\pi }{6}}}{2}+\frac {\left (4 \cos \left (t \right )^{2}+3 \cos \left (t \right ) \sin \left (t \right )-2\right ) {\mathrm e}^{-t}}{5}-\frac {2 \cos \left (t \right )}{5}+\frac {4 \sin \left (t \right )}{5} \]
✓ Solution by Mathematica
Time used: 0.644 (sec). Leaf size: 75
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]
\[ y(t)\to \frac {1}{20} e^{-t} \left (-5 e^{\pi /6} \theta (6 t-\pi ) \left (\sqrt {3} \cos (2 t)-\sin (2 t)\right )+16 e^t \sin (t)+6 \sin (2 t)-8 e^t \cos (t)+8 \cos (2 t)\right ) \]