7.21.4 problem 4

Internal problem ID [567]
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 : 4
Date solved : Monday, January 27, 2025 at 02:54:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.193 (sec). Leaf size: 17

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

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

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 32

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

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