00:01
Okay, we're gonna solve this differential equation using laplace transform, so i'm gonna take the laplace transform of both sides.
00:09
So i have l of y double prime plus 4 l of y prime plus 4 l of y equals l of 4.
00:26
So l of y double prime is this, okay, so s squared y minus s times y of 0 minus y prime of 0.
00:38
Both of those are 0, so i'm not gonna put anything there.
00:42
Okay, plus 4 times l of y prime and it is s y minus y of 0 plus 4 l of y, so 4 y equals l of 4, which is 4 over s.
01:05
Okay, factor the y out, so you have s squared plus 4 s plus 4.
01:13
Big y is 4 over s, so y is 4 over s and then that factors into s plus 2 squared.
01:24
Okay, so now what do we got to do is partial fractions to get that all separated.
01:31
So we have a over s plus b over s plus 2 plus c over s plus 2 squared equals 4.
01:43
Okay, so you get a times s plus 2 squared plus b s times s plus 2 plus c s squared equals 4.
02:10
Okay, so a s squared plus 4 a s plus 4 a plus b s squared plus 2 b s plus c s squared equals to 4.
02:28
Okay, so that gives us a plus b plus c equals 0 because there's no s squared term.
02:39
4 a plus 2 b equals 0 because there's no s term.
02:47
4 a equals 4.
02:50
Okay, that's good.
02:51
So now we know a and then we can get b and then we can get c.
02:55
So a is 1 so 4 plus 2 b is 0 so b is negative 2.
03:04
1 plus negative 2 plus c equals 0 so c is 1...