7.19.2 problem 28

Internal problem ID [542]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.3 (Translation and partial fractions). Problems at page 296
Problem number : 28
Date solved : Saturday, March 29, 2025 at 04:56:16 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }-6 x^{\prime }+8 x&=2 \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: 0.094 (sec). Leaf size: 14

ode:=diff(diff(x(t),t),t)-6*diff(x(t),t)+8*x(t) = 2; 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x = \frac {\left ({\mathrm e}^{2 t}-1\right )^{2}}{4} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 18

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

\[ x(t)\to \frac {1}{4} \left (e^{2 t}-1\right )^2 \]

✓ Sympy. Time used: 0.201 (sec). Leaf size: 19

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

\[ x{\left (t \right )} = \frac {e^{4 t}}{4} - \frac {e^{2 t}}{2} + \frac {1}{4} \]