7.21.11 problem 11

Internal problem ID [574]
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 : 11
Date solved : Saturday, March 29, 2025 at 04:57:02 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+6 x^{\prime }+8 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*}

✓ Maple. Time used: 0.190 (sec). Leaf size: 43

ode:=diff(diff(x(t),t),t)+6*diff(x(t),t)+8*x(t) = f(t); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x = \frac {\int _{0}^{t}f \left (\textit {\_U1} \right ) {\mathrm e}^{-2 t +2 \textit {\_U1}}d \textit {\_U1}}{2}-\frac {\int _{0}^{t}f \left (\textit {\_U1} \right ) {\mathrm e}^{-4 t +4 \textit {\_U1}}d \textit {\_U1}}{2} \]

✓ Mathematica. Time used: 0.07 (sec). Leaf size: 103

ode=D[x[t],{t,2}]+6*D[x[t],t]+8*x[t]==f[t]; 
ic={x[0]==0,Derivative[1][x][0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to e^{-4 t} \left (\int _1^t-\frac {1}{2} e^{4 K[1]} f(K[1])dK[1]+e^{2 t} \left (\int _1^t\frac {1}{2} e^{2 K[2]} f(K[2])dK[2]-\int _1^0\frac {1}{2} e^{2 K[2]} f(K[2])dK[2]\right )-\int _1^0-\frac {1}{2} e^{4 K[1]} f(K[1])dK[1]\right ) \]

✓ Sympy. Time used: 0.972 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
x = Function("x") 
f = Function("f") 
ode = Eq(-f(t) + 8*x(t) + 6*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\left (- \frac {\int f{\left (t \right )} e^{4 t}\, dt}{2} + \frac {\int \limits ^{0} f{\left (t \right )} e^{4 t}\, dt}{2}\right ) e^{- 2 t} + \frac {\int f{\left (t \right )} e^{2 t}\, dt}{2} - \frac {\int \limits ^{0} f{\left (t \right )} e^{2 t}\, dt}{2}\right ) e^{- 2 t} \]