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 : Tuesday, March 04, 2025 at 11:27:00 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*}

✓ Maple. Time used: 0.279 (sec). Leaf size: 28

ode:=diff(diff(x(t),t),t)+6*diff(x(t),t)+9*x(t) = f(t); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

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

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

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

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

✓ Sympy. Time used: 1.247 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x = Function("x") 
f = Function("f") 
ode = Eq(-f(t) + 9*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)
 

\[ x{\left (t \right )} = \left (t \left (\int f{\left (t \right )} e^{3 t}\, dt - \int \limits ^{0} f{\left (t \right )} e^{3 t}\, dt\right ) - \int t f{\left (t \right )} e^{3 t}\, dt + \int \limits ^{0} t f{\left (t \right )} e^{3 t}\, dt\right ) e^{- 3 t} \]