14.19.1 problem 2

Internal problem ID [2694]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.12, Dirac delta function. Excercises page 250
Problem number : 2
Date solved : Monday, January 27, 2025 at 06:09:30 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 y&=\delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.901 (sec). Leaf size: 32

dsolve([diff(y(t),t$2)+4*diff(y(t),t)+5*y(t)=Dirac(t-1),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = {\mathrm e}^{-2 t} \left (\cos \left (t \right )+2 \sin \left (t \right )\right )+\operatorname {Heaviside}\left (-1+t \right ) {\mathrm e}^{2-2 t} \sin \left (-1+t \right ) \]

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 33

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+5*y[t]==DiracDelta[t-1],{y[0]==1,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-2 t} \left (-e^2 \theta (t-1) \sin (1-t)+2 \sin (t)+\cos (t)\right ) \]