 Hello and welcome to an extra special screencast today on how to determine whether a given function is a solution to a differential equation. And I want to give an extra special shout out to Grand Valley State University's I-Dell department for the use of their light board because we couldn't have made this cool screencast without them. All right, today we're going to be looking at a differential equation. So our differential equation is dy dt equals y minus y squared. And we're going to be doing this in two different parts. We're going to look at two different functions and see if they satisfy the differential equation. See if it is or is not a solution. So the first function we're going to look at here is f of t equals 2 plus e to the t. Okay, looking back at our differential equation up here, this notation means dy dt means take the derivative of your function with respect to t. Now the fact that this is a y and this is an f doesn't matter, okay? It just matters that this means take the derivative. Then over here we're going to have to do our function then minus our function squared. Okay, so there's really only just a little bit of calculus involved with these problems. Okay, so our left hand side, so that's looking over here. So let's see what the derivative is of this function with respect to t. Well, this isn't too bad because it's just a sum. The derivative of the 2 goes away so our derivative is just e to the t. Okay, now let's look at the right hand side. So over here we've got y minus y squared. So that means take your function, so that's 2 plus e to the t, and then subtract, and then square your function, 2 plus e to the t squared. Okay, now we gotta do a little bit of algebra, that's all this is, no more calculus. So we've got 2 plus e to the t, and then minus 2 plus e to the t squared means multiply it by itself, okay? So then we've got 2 plus e to the t in the front minus we multiply all this part out, we're gonna end up with 4 plus 4 e to the t plus e to the 2t, okay? This is all properties of exponents, algebra, multiplying stuff out. So hopefully you can look at this and see there's no way we're gonna end up with where we started, but let's just go one more step. Distributing that negative through. So you can see that we're gonna end up with a whole bunch of things in here, and there's no way we're gonna get back to that left hand side. So this is not a solution. All right, we're gonna look at another function next. All right, so we're gonna be working with the same differential equation we did on the last one, but here we're gonna be looking at a different function. Okay, so again following the same idea, let's take a look at that left hand side. And the left hand side means we need to do the derivative of our function with respect to t. Okay, we're looking at this function, it's e to the t over 2 plus e to the t, that is a quotient, okay? So we're gonna have to use the quotient rule, so we gotta go back and remember that. So that means we wanna take our bottom function times the derivative of our top function, and then minus our top function times the derivative for our bottom function, let's see, let's try to go into there. And then all divided by that denominator squared. All right, simplifying this a little bit, so let's go ahead and multiply this numerator out. So we're gonna end up with 2e to the t plus e to the 2t. And then minus e to the 2t all over 2 plus e to the t squared. Okay, e to the 2t is canceled. So we end up with a function that says 2e to the t, all over 2 plus e to the t squared. So that's our left hand side. Now, let's see what our right hand side's gonna look like. That was the calculus, now we got the algebra. So the right hand side says I wanna take my function. So that's this guy in here, and I want to subtract it then from the same function squared. So I'm gonna have e to the t over 2 plus e to the t minus that same thing, e to the t, 2 plus e to the t squared. Now, let's look at the second piece first. So squaring this fraction means I'm gonna be squaring the numerator and the denominator separately. So I'm gonna end up with e to the 2t over 2 plus e to the t squared. Okay, hopefully you guys can all read that, okay? So looking back at our first fraction, we need to find a common denominator then. So I need to multiply this fraction then by 2 plus e to the t over 2 plus e to the t in order to get a common denominator. Okay, so distributing everything through, I'm gonna end up with 2e to the t, so multiplying that there. Miner plus e to the 2t, and then that's gonna end up being over 2 plus e to the t squared. Okay, smashing these two fractions together, you'll notice our e to the 2t's are gonna cancel. So we're gonna end up with 2e to the t, all over 2 plus e to the t squared. Which is exactly the same function we got on the left hand side. So this is a solution. Thank you for watching.