7.20.8 problem 37

Internal problem ID [562]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.4 (Derivatives, Integrals and products of transforms). Problems at page 303
Problem number : 37
Date solved : Monday, January 27, 2025 at 02:54:50 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+2 x^{\prime }+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: 26

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

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

✓ Solution by Mathematica

Time used: 0.065 (sec). Leaf size: 85

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

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