Final answer:
To find the inverse Laplace transform of (s + 17) / (s² + 4s -5), we can use partial fraction decomposition. After factoring the denominator, we write the function as a sum of two fractions. Solving the resulting system of equations, we find the values of A and B and then write the function in terms of e^t and e^(-5t).
Step-by-step explanation:
To find the inverse Laplace transform of the function (s + 17) / (s² + 4s -5), we can use partial fraction decomposition. First, we need to factor the denominator into two linear factors: (s + 5)(s - 1). Then, we can write the function as the sum of two fractions:
(s + 17) / [(s + 5)(s - 1)] = A / (s + 5) + B / (s - 1)
To determine the values of A and B, we can multiply both sides of the equation by (s + 5)(s - 1) and equate the numerators:
(s + 17) = A(s - 1) + B(s + 5)
Expanding and simplifying this equation, we get:
17 = (A + B)s + (5B - A)
We now have a system of equations:
A + B = 0 (Equation 1)
5B - A = 17 (Equation 2)
Solving this system of equations, we find that A = -6 and B = 6.
Now, we can rewrite the function using the values of A and B:
(s + 17) / [(s + 5)(s - 1)] = -6 / (s + 5) + 6 / (s - 1)
The inverse Laplace transform of -6 / (s + 5) is -6e^(-5t) and the inverse Laplace transform of 6 / (s - 1) is 6e^t. Therefore, the inverse Laplace transform of the given function is:
f(t) = -6e^(-5t) + 6e^t