20.23.7 problem Problem 7

Internal problem ID [3979]
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 7
Date solved : Monday, January 27, 2025 at 08:05:35 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 2.863 (sec). Leaf size: 30

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+5*y(t)=Dirac(t-Pi/2),y(0) = 0, D(y)(0) = 2],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.116 (sec). Leaf size: 34

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

\[ y(t)\to -e^{-t} \left (e^{\pi /2} \theta (2 t-\pi )-2\right ) \sin (t) \cos (t) \]