00:01
All right, so we're going to solve this differential equation by various sharp parameters.
00:06
So first, let me rewrite it in a way that just makes a little easier to understand.
00:12
So we're going to get the leaving terms by itself.
00:14
So it's y double prime by itself.
00:16
So we get y prime minus 2 y prime plus 2 y.
00:22
We're divided by 4.
00:26
E to x, secan, x.
00:29
All right.
00:31
So there we go.
00:32
Now we're going to look at this.
00:33
And we want to look at the left side of our equation, and we want to determine our general, or excuse me, our general solution based off of this side.
00:55
So we get m squared minus 2m plus 2 is 0.
01:06
Can't factor this is going to be a quadratic equation.
01:09
So we get m equals 2 plus minus the square root of, 4 b squared b squared minus 4 times 2 which is 8 and that's all over 2 and you reduce this and you end up with 1 plus or minus i which means the general solution is going to be see we're going to get a awesome train of thought here we're going to get a an exponential with a real part plus an imaginary part, which we can rewrite using demosphere theorem as e to the x, cosine x plus e2, e to the x, sine x.
02:24
So is that.
02:25
And then we need to find our particular solution, which is where this right side will come into play.
02:32
And we're going to let you equal e to the x cosine x and then we're going to let v equal e to the x sine x because we're supposed to use variation of parameters and so we're going to go ahead and find the ron skin in order to do that so the ron skin of and you could write e to x cosinex but it's just look how nice that is to write so it's going to be u v derivative of u and that's going to equal e to the x cosine x e to x sine of x derivatives negative x sine of x derivative of this it's e to the x cosine x boom and then we get let's see we multiply those together we get e to the 2x, cosine squared, plus we're going to subtract the product of this, but it's going to be subtracting the negative, so we kind of positive.
03:53
And look how lovely this is.
03:54
We have cosine squared and sine squared, so this is going to become, we'd factor out the e to the 2x, and then, yeah, we'd have the cosine squared plus the sine squared, but that's just one, so we'd just equal it's one, so it's just e to the 2x, which is great, and it's not equal to zero, that means we're going to get a solution.
04:14
That's good.
04:15
And we're going to use this here momentarily to find our particular solution.
04:20
Because our particular is going to be a of x of e to the x, cosine x, plus b of x, e to the x, e to the x, sine x.
04:37
And this is what we mean by variation of parameters like we're letting a because normally is we'd have just just a for what a is but we're going to have it vary so that's why we have eight is a function of x and just notice up here i'll put this in red where we have y double prime plus p of x y prime plus q of x y equals r of x this is the general form we have and that's why we have and that's we try to get y prime, y double prime by itself, because then we have this form.
05:18
And in this case, we went p of x to equal, is like your negative 2, and 2 would be q of x.
05:26
And basically trying to find the functions by those equal negative 2, 2, and then this value for r.
05:37
And now we're going to use this lovely integral here.
05:43
A of x is the integral minus v times r of x, ultimately it is r, knowing it's a function of x.
05:52
Times the ronskin respect to x and we're going to do the same thing for v of x in terms of finding the integral but it's different it's u of r or u times r pardon me let's go and work these integrals all right so this one for a we just substitute in our values so we have the integral we've got negative let's see what we say v was v was the sign.
06:32
So e to the x, sign of x, times the r, which is our one quarter, e2x, sican x, all over the ranskin.
06:54
And this can be so nice because things cancel out.
06:56
Look at this.
06:57
We got some lovely here.
06:58
That's going to cancel with, this is e to the 2x, that's going to cancel there.
07:03
And so then we get the negative integral of sign over code.
07:09
Cosine because secant is it a cosine so it's sign of a cosine and this is tangent so it's a it you look it's a table integral so we get oh we have a quarter whoops minus a quarter got remember that minus quarter minus a quarter natural log of cosine of x that's what a of x equals which is great b so we're just going to take work this down here so we get the b of x equals, so e to x cosine x, just do the substitution for r, which we said was, or u, which we said is x cosine.
08:05
Then we've got one quarter, e to x secant, because that's what we said r was equal to, all over.
08:15
You did it have two x, integrating with respect to x.
08:19
So this is going to give us, we have some lovely canceling out here, either the x, with the two at, either 2x, one quarter comes out, and cosine over cosine is just one, so it's just cosine x over cosine x, which is just one, we're just back to x...