7.18.9 problem 9

Internal problem ID [538]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.2 (Transformation of initial value problems). Problems at page 287
Problem number : 9
Date solved : Saturday, March 29, 2025 at 04:56:10 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+3 x&=1 \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: 0.091 (sec). Leaf size: 18

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+3*x(t) = 1; 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x = \frac {1}{3}-\frac {{\mathrm e}^{-t}}{2}+\frac {{\mathrm e}^{-3 t}}{6} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 28

ode=D[x[t],{t,2}]+4*D[x[t],t]+3*x[t]==1; 
ic={x[0]==0,Derivative[1][x][0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {1}{6} e^{-3 t} \left (e^t-1\right )^2 \left (2 e^t+1\right ) \]

✓ Sympy. Time used: 0.185 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(3*x(t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)) - 1,0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {1}{3} - \frac {e^{- t}}{2} + \frac {e^{- 3 t}}{6} \]