7.21.3 problem 3

Internal problem ID [566]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.6 (Impulses and Delta functions). Problems at page 324
Problem number : 3
Date solved : Monday, January 27, 2025 at 02:54:55 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+4 x&=1+\delta \left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.245 (sec). Leaf size: 31

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

\[ x \left (t \right ) = \operatorname {Heaviside}\left (t -2\right ) \left (t -2\right ) {\mathrm e}^{-2 t +4}+\frac {1}{4}+\frac {\left (-2 t -1\right ) {\mathrm e}^{-2 t}}{4} \]

✓ Solution by Mathematica

Time used: 0.101 (sec). Leaf size: 36

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

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