7.21.10 problem 10

Internal problem ID [573]
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 : 10
Date solved : Monday, January 27, 2025 at 02:54:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+6 x^{\prime }+9 x&=f \left (t \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: 0.279 (sec). Leaf size: 28

dsolve([diff(x(t),t$2)+6*diff(x(t),t)+9*x(t)=f(t),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = -\int _{0}^{t}\left (-t +\textit {\_U1} \right ) {\mathrm e}^{-3 t +3 \textit {\_U1}} f \left (\textit {\_U1} \right )d \textit {\_U1} \]

✓ Solution by Mathematica

Time used: 0.065 (sec). Leaf size: 93

DSolve[{D[x[t],{t,2}]+6*D[x[t],t]+9*x[t]==f[t],{x[0]==0,Derivative[1][x][0] ==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to e^{-3 t} \left (-t \int _1^0e^{3 K[2]} f(K[2])dK[2]+t \int _1^te^{3 K[2]} f(K[2])dK[2]+\int _1^t-e^{3 K[1]} f(K[1]) K[1]dK[1]-\int _1^0-e^{3 K[1]} f(K[1]) K[1]dK[1]\right ) \]