Internal problem ID [2384]
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 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+6 y^{\prime }+13 y-\left (\delta \left (t -\frac {\pi }{4}\right )\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 5, y^{\prime }\relax (0) = 5] \end {align*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 44
dsolve([diff(y(t),t$2)+6*diff(y(t),t)+13*y(t)=Dirac(t-Pi/4),y(0) = 5, D(y)(0) = 5],y(t), singsol=all)
\[ y \relax (t ) = \frac {\left (-\theta \left (t -\frac {\pi }{4}\right ) {\mathrm e}^{\frac {3 \pi }{4}-3 t}+10 \,{\mathrm e}^{-3 t}\right ) \cos \left (2 t \right )}{2}+10 \,{\mathrm e}^{-3 t} \sin \left (2 t \right ) \]
✓ Solution by Mathematica
Time used: 0.121 (sec). Leaf size: 121
DSolve[{y''[t]+46*y'[t]+13*y[t]==DiracDelta[t-Pi/4],{y[0]==1,y'[0]==-1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{516} e^{-2 \sqrt {129} t-23 t-\frac {\sqrt {129} \pi }{2}} \left (2 e^{\frac {\sqrt {129} \pi }{2}} \left (\left (129+11 \sqrt {129}\right ) e^{4 \sqrt {129} t}+129-11 \sqrt {129}\right )-\sqrt {129} e^{23 \pi /4} \left (e^{\sqrt {129} \pi }-e^{4 \sqrt {129} t}\right ) \theta (4 t-\pi )\right ) \\ \end{align*}