Ans) Given equation = (- 648 + 18 s2 - 6 s4 ) / ( s5 - 36 s3)
The equation can be rewritten as : (18 / s3) - (3 / {s + 6}) - (3 / {s -6})
=> L-1 [(18 / s3) - (3 / {s + 6}) - (3 / {s -6})]
Using property of Laplace transform,
L-1 [(18 / s3) - (3 / {s + 6}) - (3 / {s -6})] = L-1 (18 / s3) - L-1(3 / {s + 6}) - L-1 (3 / {s -6})
= 9 L-1 ( 2 / s3) - 3 L-1 (1 / {s + 6}) - 3 L-1 (1 / {s - 6})
We know, inverse Laplace transform of n ! / sn+1 = tn
Hence, L-1 ( 2 / s3) = t2
We know, inverse Laplace transform of 1 / (s - a) = eat
=> L-1 (1 / {s + 6}) = e-6 t
We know, inverse Laplace transform of 1 / (s - a) = eat
=> L-1 (1 / {s - 6}) = e6 t
Hence, putting values,
9 L-1 ( 2 / s3) - 3 L-1 (1 / {s + 6}) - 3 L-1 (1 / {s - 6}) = 9 t2 - 3 e-6 t - 3 e6 t
Therefore, required inverse Laplace transform ,
L-1 ((- 648 + 18 s2 - 6 s4 ) / ( s5 - 36 s3) = 9 t2 - 3 e-6 t - 3 e6 t
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