Analize the circuit response to applied voltage asociated with the function :(s + 1)/(s + 2)(s^2+1)(s^2+4)

According to the Laplace transform, the circuit responses
are associated with rational functions.

We'll decompose the
given rational function in the elementary quotients, which are arising from standard
circuits responses.

(s + 1)/(s + 2)(s^2+1)(s^2+4) = A/(s+2)
+ (Bs + C)/(s^2 + 1) + (Ds + 2E)/(s^2 + 4)

We'll calculate
A,B,C,D,E.

We'll determine LCD for the ratios from the
right side:

s+1 = A(s^2+1)(s^2+4) + (Bs + C)(s+2)(s^2 + 4)
+ (Ds + 2E)(s+2)(s^2 + 1)

We'll remove the
brackets:

s+1 = As^4 + 5As^2 + 4A + (Bs^2 + 2Bs + Cs +
2C)(s^2 + 4) + (Ds^2 + 2Ds + 2Es + 4E)(s^2 + 1)

s+1 = As^4
+ 5As^2 + 4A + Bs^4 + 4Bs^2 + 2Bs^3 + 8Bs + Cs^3 + 4Cs + 2Cs^2 + 8C + Ds^4 + Ds^2 +
2Ds^3 + 2Ds + 2Es^3 + 2Es + 4Es^2 + 4E

We'll combine like
terms:

s+1 = s^4(A + B + D) + s^3(2B + C + 2D + 2E) +
s^2(5A + 4B + 2C + D + 4E) + s(8B + 4C + 2D + 2E) + 4A + 8C +
4E

The correspondent coefficients must be
equal:

A + B + D = 0

2B + C +
2D + 2E = 0

5A + 4B + 2C + D + 4E =
0

8B + 4C + 2D + 2E = 1

4A +
8C + 4E = 1

The response
is:

Ae^-2t + Bcos t + Csin t + Dcos 2t + Esin
2t