00:01
Right, here's a first order linear differential equation.
00:05
It's non -homogeneous, and we are going to solve it using laplace transforms.
00:09
Yay, i like laplace transforms.
00:12
Okay, so we're going to just go ahead and start taking a laplace transform of our differential equations.
00:18
Let's do that.
00:20
Okay, so the derivative form is s, y of s minus the initial condition.
00:25
Then we have four times the laplace transform of y, and that is going to equal to one times what will end up being a shift in the frequency domain.
00:36
So normally i get one over s, but now instead i'm going to write one over s minus 10 because that exponential multiplier.
00:44
All right, excellent.
00:46
Okay, so now i'm going to go ahead and factor out y of s out of the two terms with it, leaving behind s plus four.
00:56
I'm going to add the seven to both sides.
00:58
So i get one over s minus 10 plus seven.
01:01
Now i'm going to divide both sides by s plus four.
01:07
So that will give me one over s minus 10 times s plus four plus seven over s plus four.
01:17
And i want to split up that first term into partial fractions.
01:22
So i'm going to split that up and then i'll be able to think backwards and solve my differential equations.
01:28
Put a plus in between those two.
01:29
All right.
01:31
So how we do that, there are multiple tricks, but i'll just go ahead and show you basically.
01:37
I can assume that there's some constant a there and assume a constant b there.
01:42
And i'm going to recreate the original, which means get a common denominator, which means the left fraction numerator, numerator gets multiplied by the right denominator, and the right numerator gets multiplied by the left denominator.
01:56
And that numerator should equal the same as the original.
02:01
So if i distribute out, i get as plus 4a plus bs minus 10b equal to 1.
02:10
So now notice that there are no s terms on the right.
02:14
So the a plus b, those have to equal zero...