Question

Find y(t) for Y(S)= k*(s+a)/[s*(s+b)] in terms of k, a and b.

Answer 1

To find y(t), decompose Y(S) into partial fractions and use standard Laplace transform pairs to get the inverse transformation: y(t) = k/b * (1 - e^{-bt}).

Step-by-step explanation:

To find the inverse Laplace transform y(t) of Y(S) = k*(s+a)/[s*(s+b)], we need to decompose the given Y(S) into partial fractions. This helps us to easily identify inverse transforms from standard Laplace transform pairs.

Given Y(S), we can decompose it as follows:

1. Factor the denominator: Since the denominator is already factored, we do not need to do anything further for this step.
2. Decompose Y(S) into partial fractions:

Y(S) = k * ((A/s) + (B/(s+b)))

By equating the coefficients, we would solve for A and B, where:

• A = k/b
• B = -k/b

Using the standard Laplace transforms:

• L{1} = 1/s
• L{e^{bt}} = 1/(s-b)

We can therefore convert each term back to the time domain:

y(t) = k/b - k/b * e^{-bt}

Therefore, the inverse Laplace transform y(t) is:

y(t) = k/b * (1 - e^{-bt})