20.23.9 problem Problem 9

Internal problem ID [3981]
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 9
Date solved : Monday, January 27, 2025 at 08:05:39 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+3 y&=\delta \left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 5.524 (sec). Leaf size: 24

dsolve([diff(y(t),t$2)+4*diff(y(t),t)+3*y(t)=Dirac(t-2),y(0) = 1, D(y)(0) = -1],y(t), singsol=all)
 

\[ y = \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-2 t +4} \sinh \left (t -2\right )+{\mathrm e}^{-t} \]

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 37

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

\[ y(t)\to \frac {1}{2} e^{2-3 t} \left (e^{2 t}-e^4\right ) \theta (t-2)+e^{-t} \]