Internal problem ID [2386]
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 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+16 y-4 \cos \left (3 t \right )-\left (\delta \left (t -\frac {\pi }{3}\right )\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 41
dsolve([diff(y(t),t$2)+16*y(t)=4*cos(3*t)+Dirac(t-Pi/3),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = -\frac {4 \cos \left (4 t \right )}{7}+\frac {\left (7 \cos \left (4 t \right ) \sqrt {3}-7 \sin \left (4 t \right )\right ) \theta \left (t -\frac {\pi }{3}\right )}{56}+\frac {4 \cos \left (3 t \right )}{7} \]
✓ Solution by Mathematica
Time used: 0.063 (sec). Leaf size: 50
DSolve[{y''[t]+16*y[t]==4*Cos[3*t]+DiracDelta[t-Pi/3],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{8} \theta (3 t-\pi ) \left (\sqrt {3} \cos (4 t)-\sin (4 t)\right )+\frac {4}{7} (\cos (3 t)-\cos (4 t)) \\ \end{align*}