20.23.8 problem Problem 8

Internal problem ID [3980]
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 8
Date solved : Monday, January 27, 2025 at 08:05:36 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+13 y&=\delta \left (t -\frac {\pi }{4}\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 5.428 (sec). Leaf size: 50

dsolve([diff(y(t),t$2)-4*diff(y(t),t)+13*y(t)=Dirac(t-Pi/4),y(0) = 3, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.196 (sec). Leaf size: 61

DSolve[{D[y[t],{t,2}]-4*D[y[t],t]+13*y[t]==DiracDelta[t-Pi/4],{y[0]==3,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{6} e^{2 t} \left (6 (3 \cos (3 t)-2 \sin (3 t))-\sqrt {2} e^{-\pi /2} \theta (12 t-3 \pi ) (\sin (3 t)+\cos (3 t))\right ) \]