7.21.8 problem 8

Internal problem ID [571]
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 : 8
Date solved : Monday, January 27, 2025 at 02:54:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.246 (sec). Leaf size: 27

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

\[ x \left (t \right ) = -\operatorname {Heaviside}\left (t -2\right ) \left (t -2\right ) {\mathrm e}^{-t +2}+3 t \,{\mathrm e}^{-t} \]

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 34

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

\[ x(t)\to e^{-t} \left (t (\theta (t)-\theta (0)+2)-e^2 (t-2) \theta (t-2)\right ) \]