Now, if people are *really* in for some nerd stuff, here are some fun math problems (not high school level):
If F(s) = s3 / (s2 + 9)2, find f(t) using Laplace transforms
Solution
s3 / (s2 + 9)2 = (As + B) / (s2 + 9)2 + (Cs + D) / (s2 + 9)
= (3): A[s / (s2 + 9)2] + B[1 / (s2 + 9)2] + C[ 1/ (s2 + 9)2] + (D / 3)(3 / (s2 + 9))
Using (1) and (2)...
L-1[s3 / (s2 + 9)2] = A((1 / (2 * 3))t sin (3t)) + B( (1/(2 * 33)) sin (3t)) - ((1 / (2 * 32)) t cost (3t))) + C cost (3t) + (D / 3) sin (3t)
multiply both sides of (3) by (s2 + 9)2 to get
s3 = As + B + (s2 + 0)(Cs + D)
S3: 1 = C,
S2: 0 = D,
S: 0 = A + 9C,
1 = s0: 0 = B + 9D.
A = -9,
B = 0,
C = 1,
D = 0.
(1): L-1[s / (ss + b2)] = (1 / 2b)t sin (bt)
(2): L-1[1 / (s2 + b2)2] = )(1 / (2b3)) sin (bt)) - ((1 / 2b2) / (t cost (bt)))
God I loved DiffEQ.
