00:01
In this problem we are going to compute the inverse laplace transform of this expression.
00:08
Exponential minus pi s over 2 divided by s, s squared plus pf.
00:18
Inverse laplace transform exponential minus pi over 2 s times 1 over s squared plus 4.
00:31
Okay, let us perform a partial fraction decomposition on this second.
00:35
Factor.
00:37
So we want to rewrite it as a over s plus b s plus c over s squared plus 4.
00:45
Then we have a s squared plus 4 plus b s plus c times s equal to 1.
00:53
On the left we have 4 a plus c times s plus a plus b times s squared equal to 1.
01:04
So if you compare the terms on the left and right we see that this coefficient must be equal to 0 this coefficient must be equal to 0 and this constant turn equal to 1 so we have a equal to 1 over 4 b equal to minus 1 over 4 and c equal to 0 okay then we have inverse lo -plus transform of exponential minus pi over 2 s we have a times 1 over s so 1 over 4 times 1 over s and we have okay c equal to zero so we have just b which is minus one over four so s over s squared plus four okay now these are all elementary functions so we remember that exponential something times s is the la plus transform of the direct delta so for the first is exponential factor we have the laplace transform of dirac delta t minus pi over 2.
02:26
And for this second factor we have the laplace transform of the following.
02:33
1 over 4 times 1 over s is the laplace transform of 1 minus we have 1 over 4.
02:44
S over s squared plus, okay 4 is 2 squared.
02:49
So this is the laplace transform of cosine 2t...