20.23.6 problem Problem 6

Internal problem ID [3978]
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 6
Date solved : Monday, January 27, 2025 at 08:05:34 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y&=\delta \left (t -3\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: 3.339 (sec). Leaf size: 23

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

\[ y = \frac {\operatorname {Heaviside}\left (t -3\right ) \sinh \left (-6+2 t \right )}{2}+\frac {\sinh \left (2 t \right )}{2} \]

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 44

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

\[ y(t)\to \frac {1}{4} e^{-2 (t+3)} \left (\left (e^{4 t}-e^{12}\right ) \theta (t-3)+e^6 \left (e^{4 t}-1\right )\right ) \]