00:04
Well, good afternoon, boys and girls.
00:07
So we have some more, we have some more differential equations which we need to solve using our favorite method here, which is, well, anyhow, it's this series method.
00:27
I'm actually forgetting his name.
00:30
But so we're going to have to solve this.
00:39
And so we're going to have two linearly independent solutions.
00:45
And we have to find at least the first three coefficients of both of them.
00:51
And i'm sort of skipping a few steps here because you should have, if you've looked at 8 .6, you should have already done a lot of the stuff already know how to do it.
01:07
Hopefully by 8 .7 you can do some of the steps in your own.
01:11
If you can't, i would suggest going back because the problem we're looking at here, we actually did earlier in 8 .6 in number 20.
01:21
So just go back and look at that if you don't know how to do it.
01:25
But anyhow, so the first thing we want is the r values, and it turns out that it's b zeros to three halves q zero is zero and the um what you might call indicial equation um implies that we get the two roots r1 to be zero and or r2 to be negative one half and of course r one minus r two is not an integer so we can apply uh seven point a theorem seven point a which tells us that we can, the forms of the solutions.
02:08
And that becomes straightforward.
02:11
So now we plug w of rx into it.
02:15
Sorry about that.
02:17
Made a mistake here.
02:19
Okay, so we plug w of rx into it.
02:23
And we do the algebra on it.
02:29
So looks like this.
02:31
This and just a word in case you can't see it very well.
02:38
What i have here is i expanded the number two, equation number two out.
02:49
So now i have each term expanded out here, each of the five terms.
02:55
And then i wrote down each term explicitly.
03:02
So in particular, this is that term.
03:08
This one corresponds to that, so on and so forth, all the way down the line.
03:16
And of course, we know it's equal to zero.
03:22
And i think, yeah, right here, what i did is i did a little more simplifying.
03:27
So now i sort of gathered these guys up into such a way.
03:33
The powers of xes are matching in each line.
03:37
So this line has all the x to the n plus r powers, and this line has the n minus one plus r powers.
03:48
Just to line them up, because in the next step, what we're going to do is we're going to just do the simplifying a little bit more, and voila, we simplify it down...