Find inverse Laplace transform of (s)/(s^2+s+1)

trkalo84 2022-09-26 Answered
Solve the following equation:
y + y + y = 1 + e t
After applying the Laplace transform and using partial fractions I have terms of the form:
1 s 2 + s + 1
and
s s 2 + s + 1
which I don't know how to reverse transform.
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Answers (1)

r2t1orrso
Answered 2022-09-27 Author has 8 answers
You didn't give the initial conditions...a bit hard to check your solution
You have that
s s 2 + s + 1 = s ( s 2 + s + 1 / 4 + 3 / 4 ) = s ( s + 1 / 2 ) 2 + 3 / 4
s ( s + 1 / 2 ) 2 + 3 / 4 = s + 1 / 2 ( s + 1 / 2 ) 2 + 3 / 4 1 / 2 ( s + 1 / 2 ) 2 + 3 / 4
And look at the sine and cosine function in the table...
L ( e a t cos ( β t ) ) = s a ( s a ) 2 + β 2
L ( e a t sin ( β t ) ) = β ( s a ) 2 + β 2
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