Question

0 Let's add anotheu ut t redefine Let u(t)be ter fu nction hat turns off ot nlt) be a ram function that turns f a Fin1 XLt) for 20 0 Alt-リ A(3-)

A= [2    1]
    [-8 -4]

please help me find the peice wise x(t)

Answer 1

f₁₊(t,a) = {te^at , where t>0 and 0 where t<0, f_1-(t,a) = 0 where t>0 and -te^at t<0

The function f₁₊(t) and f_1-(t) are continue. Due to the function $\varphi$_f (T,s,t) takes the form

$\varphi$_f(T,s,t) = $\partial$ e^{(a - s)t} /  $\partial$a 1- e^{(a - s)T}, 0 $\leq$ t $\leq$ T

or, in expanded form,

$\varphi$_f(T,s,t) = e^{(a - s)} t[1 - e^{(a - s)T}] + Te^{(a - s)T} / [1 - e^{(a - s)T}]², 0 $\leq$ t $\leq$ T.

It can be immediately verified that $\varphi$_f(T, s, +0) = $\varphi$_f(T, s, -0) in accordance with general theory. The corresponding function

$\varphi$_f(T, s, t) can be calculated by periodic continuation for the argument t. For a = 0 we have F(s) = s² , and form it followed that,

$\varphi$_f(T,s,t) = e^-st (1 - e^-sT)t + Te^-sT / (1 - e^-sT)² ,  0 $\leq$ t $\leq$ T

For Re s> Re a and  0 $\leq$ t $\leq$ T the function coincides with  $\varphi$₁₊(T,s,t), and for Re s < Re a it coincides with $\varphi$_f- ( T,s,t),