00:01
All right, let's take a look at solving this differential equation.
00:06
I'm going to start by looking at the homogeneous case.
00:10
All right, so we're basically going to start by looking at t cubed y prime plus 4 t squared y equals zero.
00:20
Our general solution will be made up of the homogeneous plus our particular solution.
00:25
Okay, so this is of the form where i can let y equals t to the m power.
00:33
So y prime will be m, t to the m minus one.
00:37
Y double prime is m times m minus one, t to the m minus two.
00:43
So i'm going to go ahead and substitute in for my y prime and my y.
00:48
Oh, i did more derivatives than i needed to.
00:51
It's only first order, not second order.
00:53
Oh, so there we go.
00:54
So that makes it even easier.
00:57
Okay, so let's sub in.
00:58
I will get tq times y prime m, t to the m minus 1, plus 4 t squared, and then y is t to the m.
01:08
So if you look carefully, every term has a t to the m plus two.
01:13
So i can't see what's left behind.
01:16
I'll get m plus four.
01:19
Okay, so therefore, m equals minus four.
01:23
Excellent.
01:23
So that is my particular solution.
01:27
So far i know that y homogeneous is c1e to the might.
01:32
Whoops, not e.
01:34
It's t.
01:35
So let's make it t.
01:36
So c1t to the minus four.
01:40
All right.
01:40
So we have one of our solutions.
01:43
But now we need to get our particular solutions.
01:46
And i'm going to go ahead and do it by assuming that our second solution, our particular solution is of the form u times are homogeneous and i'll do it without the constant i'll put it back in at the end so i'll get u times um uh t to the minus four so maybe what i'll do is i'll just say y1 is t to the minus 4 and that way i can just be consistent here to not worry about the c so i'll say u y1 and then that is then u or u itself as a function of t and y1 is t to the minus four.
02:31
And if i want, i can write that as, well, leave it like that for now.
02:36
I'll clean it up after.
02:38
Okay.
02:39
So now it's fine y2 prime.
02:42
That will be product rule.
02:43
So i'll get u prime, t to the minus four, minus four, t to the minus five, u.
02:50
So that i can clean up as, let's see, u prime over t to the fourth minus four u over t over t to the fifth.
03:04
Okay, so now i'm going to sub in and see what happens.
03:07
So let's do that.
03:09
So i'm going to sub into my, now my non -homogeneous differential equation, see what happens.
03:16
Okay, when i do that, i'll get t cubed times the derivative.
03:21
So i'll get u prime over t to the fourth minus four u over t to the fifth.
03:28
Then i'm going to do plus four t squared times y.
03:31
And y is you...