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 : Tuesday, March 04, 2025 at 11:26:43 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*}

✓ Maple. Time used: 0.279 (sec). Leaf size: 26

ode:=diff(diff(x(t),t),t)+2*diff(x(t),t)+x(t) = f(t); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

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

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

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

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

✓ Sympy. Time used: 1.218 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x = Function("x") 
f = Function("f") 
ode = Eq(-f(t) + x(t) + 2*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)
 

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