20.23.10 problem Problem 10

Internal problem ID [3982]
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
Date solved : Monday, January 27, 2025 at 08:05:40 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 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 )&=5\\ y^{\prime }\left (0\right )&=5 \end{align*}

✓ Solution by Maple

Time used: 3.148 (sec). Leaf size: 42

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)
 

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

✓ Solution by Mathematica

Time used: 0.280 (sec). Leaf size: 121

DSolve[{D[y[t],{t,2}]+46*D[y[t],t]+13*y[t]==DiracDelta[t-Pi/4],{y[0]==1,Derivative[1][y][0] ==-1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{516} e^{-2 \sqrt {129} t-23 t-\frac {\sqrt {129} \pi }{2}} \left (2 e^{\frac {\sqrt {129} \pi }{2}} \left (\left (129+11 \sqrt {129}\right ) e^{4 \sqrt {129} t}+129-11 \sqrt {129}\right )-\sqrt {129} e^{23 \pi /4} \left (e^{\sqrt {129} \pi }-e^{4 \sqrt {129} t}\right ) \theta (4 t-\pi )\right ) \]