What is the relationship between poles and the form of system response?

[theone]

Homework Statement

I am trying to understand what is meant by "poles can be used to obtain the form of the system response''

This is an example for a transfer function (s+2)/(s+5) and a step function input
http://postimg.org/image/97vnos9uz/

Homework Equations

The Attempt at a Solution

For example, the input pole of s=0 generates the form of the forced response.
I don't understand what s=0 has to do with the output transform (2/5)/s and the time response of 2/5

 

[vela]

Recall when you solved differential equations like $y'' + 5y' + 6 = 0$. By substituting a solution of the form $y=e^{rt}$, you obtained the characteristic equation $r^2 + 5r + 6 = 0$. After you obtain the roots, you could write down that the solution to the differential equation was $c_1 e^{-2t} + c_2 e^{-3t}$.

The transfer function is the impulse response of the system. That is, it's the solution to $y'' + 5y' + 6 = \delta(t)$. If you take the Laplace transform of both sides, you end up with $(s^2+5s+6)Y(s) = 1$. Solving for Y(s), you get
$$Y(s) = \frac{1}{s^2+5s+6}.$$ Note that the denominator is identical to the characteristic polynomial, so if you know poles of the transfer function, which are the roots of the characteristic polynomial, you know what terms are going to show up in the homogeneous part of the system response.