Determining the domain of particular solution to differential equation

0:00- [Instructor] We're told a particular solution passing through the point five, two to the differential equation, this stuff, is equal to y is equal to 1/2 times the square route of x squared minus nine, or that this right over here is the particular solution.

0:15They want us to figure out the domain of this particular solution. So pause this video, have a go at that. All right, so if I weren't looking at this differential equation here, and if I were to just look at this equation, y is equal to 1/2 times the square route of x squared minus nine.

0:31The graph would look something like this. So let me just draw a rough version of it. So this is the x-axis, this is the y-axis, this is positive three. This is, let me make sure it's okay. That looks well, maybe a little bit further. This is negative three. The graph is going to look something like this.

0:56I can take... So it's gonna look like that. It's going to look something like this. And we have the point five, two. So that would be five, two would put us right over there. This is five, this is the point five, two. So if you were to just say, what's the domain of this thing here, you can really just take any x value that keeps what we have under the radical from becoming less than zero.

1:31It can be equal to zero because you could take the square root of zero. So if we were just looking at this equation, we would say that domain is all Xs that are less than or equal to negative three, or all the Xs that are greater than or equal to three. We can see that over here.

1:48Now, we're going to restrict this more because we aren't just dealing with this equation. We are dealing with that as a particular solution to a differential equation. So the first thing we need to realize is if we are going through, this is a particular solution to the point going through five, two, or going through the point five, two.

2:09So if this is a solution, maybe you're trying to model some phenomena, maybe it's some type of position function or something else. Well, you can never get to this other curve right over here. You could move in that direction over there, you could move in that direction there, or maybe you start over here and then you move in that direction, you go through five, two, but there's no way that you can get to this other branch.

2:32There is a jump between these two. And so we'd immediately restrict that part of the domain away, because you can't get from this point somehow to that side of the curve. And so I'll restrict it now to just x is greater than or equal to three, but we're gonna have to restrict it even further, because if x is greater than three, we get nice positive values under the radical, everything works out well.

3:02Y over here is going to be a positive value. But if x is equal to three, you get nine minus nine, which is zero times 1/2, you get y equals zero. Well, y equals zero would get us a zero in the denominator right over here in the differential equation. So our differential equation can't be defined there.

3:21So we're gonna say that x cannot be equal to three, because that gets us y equals zero. And so the real domain of this particular solution is that x has to be greater than three. Or if we wanna write it in interval notation, we go between three, not including three, that's why I have a parentheses here instead of a bracket, all the way to infinity.

3:44You could express the domain either way.