Use appropriate algebra and theorem 7.2.1 to find the given inverse laplace transform. (write your answer as a function of t.) ℒ−1 s s2 + 3s − 4

We assume you want to find the inverse transform of s/(s^2 +3s -4). This can be written in partial fraction form as

(4/5)/(s+4) + (1/5)/(s-1)

which can be found in a table of transforms to be the transform of

(4/5)e^(-4t) + (1/5)e^t

There are a number of ways to determine the partial fractions. They all start with factoring the denominator.

s^2 +3x -4 = (s+4)(s-1)

After that, you can postulate the final form and determine the values of the coefficients that make it so. For example:

A/(s+4) + B/(s-1) = ((A+B)s + (4B-A))/(s^2 +3x -4)

This gives rise to two equations:

(A+B) = 1

(4B-A) = 0

225x²+4=0

225x²=0-4

225x²=-4

x²=-4/225

x=±√-4/225