00:03
Okay, this has a part a.
00:07
Show that the functions f1 of x equals e to the r1 of x.
00:24
F2 of x equals e to the r sub 2 of x, and f2 of x equals e to the r sub 2 of x, and f sub 3 of x equals e to the r sub 3 of x have a ronskian that's shown in the book well let's go ahead and do it and then we will see what is shown we need to write down the functions in the first row now derivatives are 1 e to the r1x, r1 squared e to the r1x.
01:26
R2e to the r2x, r2 squared e to the r2x, r2 squared e to the r2x.
01:38
R3e to the r3x, r3 squared e to the r3x.
01:50
Okay, very good.
01:54
Now, so this is the determinant.
02:06
But let's keep going with this.
02:16
If we take the determinant, then the way i like to take the determinant is i like to write the first two rows over again.
02:39
Wait a minute.
02:44
Before we do that, thinking about another way to do this.
02:49
Let's just say that this is a matrix, because it is.
02:54
Now, we can take...
03:01
No, that doesn't work.
03:03
I was going to factor out an e to the r from each of them.
03:07
So, no, never mind.
03:10
Let's go back to the original thought.
03:15
Let's go ahead and do the determinant.
03:22
That's going to equal those three.
03:26
Oh, i forgot.
03:27
I was going to write the first two columns again.
04:17
Okay, in calculating the determinant, we've got all of these that would be i'm going to write one times r2 times r3 squared times e to the r1x times e to the r2x times e to the r3x that would be e to the r1 because when when exponents are multiplied together, you can add the powers x.
05:11
Now let's look at the next term.
05:13
It's going to be 1 times r3 times r1 squared.
05:23
And again, we're going to have e to the r2, r3, r1.
05:32
That's the same as r1, r2, r3 using the associative property, x power, plus this is of the same form again.
05:52
1, r1, r2 squared.
06:05
And again, we get e to the r1 plus r2 plus r3x power.
06:23
And then it would continue, we're going to do minus.
06:27
This minus this minus this minus this is going to be this this and they're all going to be of the same form and so you notice that in the determinant the e to the r1 plus r2 plus r3 power x term didn't leave enough room this factors out and and then the remaining terms are just the determinant without writing the ease.
07:38
So the remaining terms are one, if you look here first, 1r2, r3 squared.
07:56
1 are 3 squared.
08:11
1 r1 r2 squared all right and that's what we're supposed to show now we need to also show that that simplifies to something so let's go ahead and show that i'm just going to come down here and i'm going to do 1 r1 squared 1 are 2 squared, 1, r2 squared, 1, r3, r3 squared.
09:04
I accidentally had a square in there.
09:17
Again, i like to write 1, r1, r1 squared, 1, r2, 2 squared, and that equals, let's do these 3, 1.
09:40
Actually, i'm not even going to write the 1.
09:45
R2r3 squared plus r3r1 squared plus r1r2 squared minus these three.
10:13
R2 r1 squared minus r3r2 squared minus r1r3 squared.
10:30
Minus r1r3 squared.
10:54
I'm trying to scroll down.
10:56
Here we go.
10:57
Now, we're trying to show, according to the textbook, that that's the same as r3 minus r1, r3 minus r2, r2 minus r1.
11:25
So let's go ahead and just do the foil method with the first two terms here.
11:29
That's going to give us r3 squared.
11:34
Outer is going to be minus r2, r3.
11:38
Inner will be minus r1, r3, and last will be plus r2 squared.
11:46
And we have to take that times r2 minus r1.
11:55
So scrolling down.
11:58
R2 times everything.
12:00
That would be r2.
12:01
I don't need the parentheses, by the way.
12:08
R3 squared minus r3 r2 squared minus r1 r2 squared minus r1 r2 r3 plus r2 cubed minus r1 times everything r1 r3 squared minus r1 r2 r3 3 3 minus r3, no wait, i'm getting my pluses and minus is mixed up.
13:19
The second one in negative times negative makes a positive.
13:24
The third one in negative times negative makes a positive.
13:28
That's going to be r1, r3, r1 squared, minus r1r2 squared.
13:50
Okay.
13:56
Those canceled out.
13:58
And so r2r3 squared is the first term here, and it's the first term here.
14:10
Minus r3r2 squared.
14:16
I don't know what to say about that r2 cubed.
14:20
Minus r1r3 squared plus r3 r1 squared.
14:36
Minus r1r2 squared.
14:43
Something's not right there.
14:46
There.
14:51
Oh, okay.
14:53
I see.
14:55
First, outer, inner, last.
14:58
My last term up here was not r2 squared.
15:06
It should have been r1r2.
15:12
And so now that's going to affect this term and this term.
15:22
Okay, now that makes more sense.
15:25
All right.
15:26
So, this will be r1r2 squared, times this, and then the last one will be minus r2r1 squared.
15:44
Now i think we're good.
15:50
R1r2 squared minus r2r1 squared.
15:57
Yes, very good.
16:00
And so, this is e to the r1 plus r2 plus r3, x.
16:17
X power times we figured out the determinant is this r3 minus r1 r3 minus r2 are 2 r whoopsies 2 minus r1 that's not r1 just barely all right and hence determine the conditions on r1 on r2 and r3 such that a set of the three functions is linearly independent on every interval.
17:19
So we would need to set that this equals zero.
17:30
However, the first part of it, e to the zero power is one, e to any power is not going to be zero.
17:49
So that portion we don't have to be concerned about.
17:55
So it's just the last portion...