 Hello, and welcome to a screencast today about solving a separable differential equation. Alright, today we are going to be finding all functions that are solutions to the differential equation, dy dt equals negative 0.2y. Okay, first thing you want to notice with this particular equation is that it is separable. Now how do we know that? Because I can actually move this y over to the other side and have this side of the equation be a function of y times dy dt and then equals something else. Okay, so let's do that. So we need to undo this multiplication then with a division. So I'm going to rewrite this as 1 over y dy dt equals negative 0.2. Okay, now what that allows us to do now is we want to go ahead and integrate both sides with respect to, hmm, which variable? That's the hard part. So here, actually it's not. If we take a look at this denominator, we know we want to integrate both sides with respect to t. Okay, because that's going to be our dependent variable for this particular problem, or independent, sorry. Okay, so let's go ahead and integrate both sides. I'll rewrite that with respect to t. So now we got functions that we can integrate. Oh, what's happening on this left-hand side? That's kind of funky, right? Okay, well you notice these dt's kind of cancel out, I guess you could say. And now we just have a swap of variables. So now I'm actually integrating this side of the equation with respect to y and this side of the equation with respect to t. How about that? Okay, so hopefully these are functions you know how to integrate. So 1 over y, the integral or the anti-derivative is the natural log of y. And then on the right-hand side, the anti-derivative of just a constant is going to be negative 0.2t. And then plus an extra constant we're going to call c. Okay, now y didn't put a c on this side of the equation because both of these are families of anti-derivatives. Okay, the reason why I didn't put that on this side of the equation is because we're going to be solving for y with respect to t. Okay, so now here, how do I undo a natural log function? Well remember that's a base of e, so I'm going to do e to both sides. So that's going to give me y equals e to the negative 0.2t plus c. Okay, and we could actually leave it like this but typically you don't want to do that. So now we've got to remember some properties of exponents. Okay, so here, hopefully you remember going backwards if you add in your exponents. That means we were actually multiplying with our bases. So we've got e to the negative 0.2t times e to the c. But e to the c, we can just call that another constant. So our final answer here is going to give us then. You want to call that constant a, another c, it doesn't really matter. So I'll just call it a e to the negative 0.2t. And that is my solution then, my family of solutions to this differential equation. If I had an initial condition, I could actually solve for my value of a. But for now, we did exactly what the problem asked for. Thank you for watching.