Find the transform of f(t):
f (t) = 3t + 2t2
Solution:
F(s) = 3/s2 + 4/s2
Find the inverse transform of F(s):
F(s) = 3 / (s2 + s - 6)
Solution:
In order to find the inverse transform, we need to change the s domain function to a simpler form:
F(s) = 3 / (s2 + s - 6) = 3 / [(s-2)(s+3)] = a / (s-2) + b / (s+3)
[a(s+3) + b(s-2)] / [(s-2)(s+3)] = 3 / [(s-2)(s+3)]
a(s+3) + b(s-2) = 3
To find a and b, we get 2 equations - one of the s coefficients and second of the rest:
(a+b)s + 3a-2b = 3
a+b = 0 , 3a-2b = 3
a = 3 , b = -3
F(s) = 3 / (s-2) - 3 / (s+3)
Now F(s) can be transformed easily by using the transforms table for exponent function:
f (t) = 3e2t - 3e-3t
