00:01
Now, in this case, we're going to take a look at, let's see here, looking at a bunch of different equations, and i'm going to do all these next few problems together because they're all very similar.
00:16
So we're given some differential equations, and i suppose i should have written them down for each problem.
00:22
So let me actually, so 36, we have s double prime plus 4s prime plus c, s equals 0.
00:34
And again, this is, notation is getting bad because, again, in engineering, this would be c and this would be k.
00:44
So, but they use b and c here for a generic differential equation.
00:50
So we'll keep with their notation.
00:55
So basically they ask for what, find the values of c that make the general solution, over -dempt, under -dempt, and critical -dempt.
01:02
Well, what you need to look at is the discriminant.
01:10
Is that the discriminant? anyway, yeah, i think so.
01:14
The thing under the square root in the quadratic formula.
01:17
So b squared minus 4c, these all have a in the characteristic equation of 1.
01:25
So a here is just 1.
01:27
So b squared minus 4c.
01:29
In this case, we have b is 4.
01:33
C is c.
01:36
And so we have 16 minus 4c.
01:39
Is that greater than equal to or less than zero? so if it's over -damped, it's going to be greater than zero.
01:48
Well, let's start.
01:50
Over -damped, it is going to be greater than zero.
01:57
Why did i...
01:59
Oh, i think i looked at them.
02:01
These are probably all messed up.
02:04
I thought they said over -damped, under -damped first.
02:08
So if it's less than zero, it is underdamp.
02:14
So that should be that.
02:15
No, i had it ready, i think.
02:18
Yeah.
02:19
So i see one down here.
02:21
That is definitely not right because i, oh no.
02:24
This is a hat.
02:26
Anyway, i think i had it right in the first place.
02:29
So if this discriminant is positive, meaning that that means we'll have real roots of our characteristic equation, which means that, well, if the system, if it's a mechanical system, we're going to assume that it's going to be underdamped.
02:45
And so we'll see that, you know, the roots are all going to be, all going to be negative.
02:55
So, let's see here.
02:57
Just because you're going to have, you know, negative b plus or minus, you know, something that is going to be less than b.
03:05
So we have this thing.
03:10
If it needs to, for it to be over -damped, we need, let me just do it on a flight.
03:18
If it's over -damped, we need this to be real numbers, or this to be positive.
03:24
So we need our characteristic groups to be positive.
03:26
So this needs to be real numbers.
03:29
So this thing needs to be positive.
03:32
So that means c needs to be less than four.
03:40
Now for it to be under damp, this needs to be negative, so because you get complex roots of our character's situation.
03:53
So for that to be negative, c needs to be greater than four.
04:05
Yeah, yeah, yeah.
04:07
I'm getting confused because i'm thinking c is the damping coefficient...