 In this video, I want to do some more examples of solving differential equations using the technique of separation of variables. This equation is a separable differential equation. You can see that in the following way. We have y times dy over dx equals x squared. Times in both sides by dx, we get y dy equals x squared dx. Fairly simple to solve it there. If we integrate both sides, the left with respect to y and the right with respect to x, we're going to get y squared over 2 equals x cubed over 3 plus a constant. And if you want to, you can solve, you can times both sides by 2, and you're going to get y squared equals two-thirds x cubed plus c. And I want to mention that at this moment, you could actually stop right here. And we could be like, hey, this is the solution to the differential equation. Is it a function exactly? Well, not really because y equals, well, we don't have y equals. We could press forward and find it y equals plus or minus the square root of two-thirds x cubed plus a constant. That would be what y equals, if you solve for it, would look like. But you'll notice that we don't get, we don't get a curve that solves the, that passes the vertical line test. It's not a function, it's usual sense. And so when we said earlier that the solution to a differential equation is a function, it's a little bit of a lie, right? So I'll apologize for that. The solution to a differential equation is actually going to be a curve, a curve in the x, y plane. And so because we don't necessarily need a function, we want a curve, because if we have the equation of a curve, we can take the derivative implicitly. Nothing has to be explicit here. Implicit differentiation, implicit anti-differentiation are perfectly appropriate in this setting. So I want you to be aware that a solution like y squared equals two-thirds x cubed plus one in some regard is more preferable than this situation where we have like these two cases plus or minus. And that problem can get further exacerbated like in this example right here. We clearly have a separable differential equation. If you separate the variables, you'll get 2y plus cosine of y dy. This equals 6x squared dx. Integrate the left-hand side with respect to y, the right-hand side with respect to x. On the left-hand side, you're going to get y squared plus sine y, sine of y equals, on the right-hand side, we're going to get 2x cubed plus a constant. And honestly speaking, that's about as far as we can get. If we try to solve for y explicitly, we ain't going to do it, right? What can we do to solve for y? I mean, we can try to get some numerical calculation here, but we have to be somewhat satisfied with what we see here. And so in these two examples, we've seen that the solution actually could be left in implicit form or like in this example, we really can't get an explicit solution. We have to be content with this implicit solution right here. The solution of a differential equation might not be explicit. That's actually quite common when you solve an equation using the separation of variables. That is, it's quite common for separable differential equations to have no explicit solution. And so in particular, as we're doing this implicit anti-differentiation, don't be against us if we don't have an explicit solution. We can be okay with a curve in the plane that's given by this implicit relationship between x and y. That's perfectly good in this situation.