Question

the forced response of a system has its laplace transform as ( )2 2 1 5 ( ) 3 25 y s f s s = ⋅ . use the partial fraction method to find its time-domain response function yf(t).

Answer 1

To find the time-domain response function, we need to use the partial fraction method. The given Laplace transform expression is (s^2 + 2s + 1)/(s^3 + 25). By factoring the denominator and equating coefficients, we find the values of A, B, and C. Taking the inverse Laplace transform, we obtain the time-domain response function y(t).

Step-by-step explanation:

To find the time-domain response function, we need to use the partial fraction method. The given Laplace transform expression is (s^2 + 2s + 1)/(s^3 + 25). To factorize the denominator, we can use the following identity: (a^3 + b^3) = (a + b)(a^2 - ab + b^2). In this case, a = s and b = 5. Factoring the denominator gives (s + 5)(s^2 - 5s + 25). So, we have:

(s^2 + 2s + 1)/(s^3 + 25) = A/(s + 5) + (Bs + C)/(s^2 - 5s + 25)

To find the values of A, B, and C, we can multiply both sides by the denominator and equate coefficients of like terms. Solving the resulting equations, we find A = 1/10, B = 1/10, and C = -1/10.

Now we can rewrite the expression as:

(s^2 + 2s + 1)/(s^3 + 25) = 1/(10(s + 5)) + (s + 1)/(10(s^2 - 5s + 25))

Taking the inverse Laplace transform of each term, we get:

y(t) = 1/10e^(-5t) + (1/10)cos(5t - π/3)e^(5t/2)sin((sqrt(3)/2)t)