20.23.13 problem Problem 13

Internal problem ID [3985]
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
Date solved : Monday, January 27, 2025 at 08:05:44 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=4 \sin \left (t \right )+\delta \left (t -\frac {\pi }{6}\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 5.585 (sec). Leaf size: 69

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)
 

\[ y = -\frac {\operatorname {Heaviside}\left (t -\frac {\pi }{6}\right ) \left (\sqrt {3}\, \cos \left (t \right )^{2}-\cos \left (t \right ) \sin \left (t \right )-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{-t +\frac {\pi }{6}}}{2}+\frac {\left (4 \cos \left (t \right )^{2}+3 \cos \left (t \right ) \sin \left (t \right )-2\right ) {\mathrm e}^{-t}}{5}-\frac {2 \cos \left (t \right )}{5}+\frac {4 \sin \left (t \right )}{5} \]

✓ Solution by Mathematica

Time used: 0.613 (sec). Leaf size: 75

DSolve[{D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==4*Sin[t]+DiracDelta[t-Pi/6],{y[0]==0,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{20} e^{-t} \left (-5 e^{\pi /6} \theta (6 t-\pi ) \left (\sqrt {3} \cos (2 t)-\sin (2 t)\right )+16 e^t \sin (t)+6 \sin (2 t)-8 e^t \cos (t)+8 \cos (2 t)\right ) \]