 One of the analytically solvable forms of differential equations is known as an exact differential equation. To understand exact differential equations, let's go back a little bit. Given any relationship between two variables, we can use implicit differentiation to produce a differential equation. For example, suppose I want to write a differential equation corresponding to the relationship x squared sine y plus y equals 25. If I differentiate with respect to x, I get, and I can rearrange this to get a differential equation of the form. Conversely, if I run into this differential equation in a dark alley one night, despite its rather terrifying form as a non-linear first-order differential equation, I happen to know that it came from x squared sine y plus y equals 25. Which means that this is a relationship that solves the differential equation. Let's think about this a little more. In single variable calculus, we viewed the relationship x squared sine y plus y equals 25 as defining y as an implicit function of x. But a different way of looking at this comes from multivariable calculus. In multivariable calculus, we viewed the relationship x squared sine y plus y equals 25 as defining a level curve of the function f of xy equals x squared sine y plus y. In other words, it's a curve defined by this function equal to a specific constant, in this case 25. It will be convenient to adopt this viewpoint. So consider a level curve of the form f of xy equals some constant c. The total derivative of f with respect to x is going to be the partial derivative of f with respect to x, plus the partial derivative of f with respect to y times the derivative of y with respect to x. Now if you haven't seen this before, the useful thing to keep in mind is this is essentially implicit differentiation using partial derivative notation. Now here's the useful thing. Even though this looks like it involves partial derivatives, it's still an ordinary differential equation because it corresponds to differential equations like dy dx, x squared cosine y plus 1, plus 2x sine y equals 0, which is the one we got before. So why is this useful? This allows us to construct a differential equation from a function of two variables by finding partial derivatives, or, working backwards, we can find the function of two variables by integration. So let's try to solve our horrifying differential equation. So we might begin with the idea suppose the solutions correspond to the level curves of f of xy equals some constant. Then our total differential must equal 0. Now equals means replaceable, so our differential equation, which is equal to 0, and our total derivative, which is also equal to 0, can be said equal to each other. So if we compare our equations, we see we have dy dx on both sides. On the left hand side it's multiplied by x squared cosine y plus 1. On the right it's multiplied by the partial of f with respect to y. But if the two sides are equal, then it must be that the partial of f with respect to y is x squared cosine y plus 1. Meanwhile, we also have the leftovers. Well, the leftovers 2x sine y must be the partial of f with respect to x. We can find f by anti-differentiating, and we have two possibilities. First, since this is the partial of f with respect to y, we can anti-differentiate with respect to y, and that gives us, or since this is the partial of f with respect to x, we can anti-differentiate with respect to x and obtain. Now we have these two constants, so let's think about this a moment. We got 2x sine y by differentiating with respect to x, and so that means this c1 could be a constant, but more generally it's a constant with respect to x. But this means it could be a function of y only. In other words, if this c1y was a function of y only, then when we differentiate with respect to x, this term vanishes. Similarly, we got this by differentiating our function with respect to y, which means that c2 must be a constant or a function of x only. And here's the important thing, the two functions have to be the same, and so we know that x squared sine y plus c1y must be x squared sine y plus y plus c2x. And that means that c1y must be y plus c2x. But wait, remember, since c1y must be a function of y only, that means c2x can only be a constant. And so c1y must be y plus a constant. And so the level curves of f of xy equals x squared sine y plus y plus a constant correspond to a solution to our differential equation. Now, since the actual solution will be of the form f of xy equals sum constant, we can omit this constant of anti-differentiation for our equation for f of xy. And so we can just say that capital F of xy must be x squared sine y plus y. Now, a good rule is that our final answer should not include any new functions or variables. And so while we do have all the necessary information, the solutions correspond to the level curves of f of xy equals c, where f of xy is equal to this function. It's better if we write the solution without any reference to this capital F function at all. And so we can just say that the general solution is x squared sine y plus y equal to c. And this leads to the following definition. An exact differential equation is an equation of the form g of xy y prime plus h of xy equals zero, where there is some function f of xy, where the partial of f with respect to y is our g function, and the partial of f with respect to x is our h function. In that case, the solutions correspond to the level curves f of xy equals constant.