Internal problem ID [5954]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 7 THE LAPLACE TRANSFORM. EXERCISES 7.5. Page 315
Problem number: 11.
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 (-\pi +t \right )\right )-\left (\delta \left (t -3 \pi \right )\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 1, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 61
dsolve([diff(y(t),t$2)+4*diff(y(t),t)+13*y(t)=Dirac(t-Pi)+Dirac(t-3*Pi),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = -\frac {\theta \left (-\pi +t \right ) \sin \left (3 t \right ) {\mathrm e}^{2 \pi -2 t}}{3}-\frac {\sin \left (3 t \right ) {\mathrm e}^{6 \pi -2 t} \theta \left (t -3 \pi \right )}{3}+\left (\cos \left (3 t \right )+\frac {2 \sin \left (3 t \right )}{3}\right ) {\mathrm e}^{-2 t} \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 53
DSolve[{y''[t]+4*y'[t]+13*y[t]==DiracDelta[t-Pi]+DiracDelta[t-3*Pi],{y[0]==1,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{3} e^{-2 t} \left (3 \cos (3 t)-\left (e^{6 \pi } \theta (t-3 \pi )+e^{2 \pi } \theta (t-\pi )-2\right ) \sin (3 t)\right ) \\ \end{align*}