00:01
In this problem we are going to evaluate the inverse laplace transform of the following expression.
00:10
2 s plus 17 divided by s s squared plus 6 s plus 34.
00:19
Okay first that has performed a partial fraction decomposition.
00:24
And we are going to note that this second factor in the denominator doesn't factorize nicely.
00:31
So we are going to just leave it as is.
00:34
So with that, we want to be able to rewrite this as a over s plus b s plus c over s squared plus 6 s plus 34.
00:48
Then we have a s squared plus 6 s plus 34 plus b s plus c times s equal to 2 s plus 17.
01:01
On the left we have 34a plus 6a plus c s plus a plus a plus b as squared.
01:19
This must be equal to 2 s plus 17.
01:23
So if we compare this equation side by side, we see that this coefficient must be equal to 0, this coefficient must be equal to 2 and this constant term must be equal to 17 which gives us a equal to one half b equal to minus one half and c equal to minus one one so we have inverse lap plus transform of one half one over s i'm going to write this term by splitting it into two so this turn plus this turn so we have minus one half s over s squared plus 6 s plus 34 minus 1 over s squared plus 6 s plus 34 now let us note that this expression in the denominator can be written as s squared plus 6 s plus 9 plus 25 which is equal to s plus 3 squared plus 5 squared.
02:46
Okay, now we want to obtain this shift in the denominator and the numerator as well.
02:53
So we will add 3 and subtract 3.
02:59
Similarly, for this last term we have s plus 3 squared plus 5 squared here...