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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
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