 The easiest types of differential equations to solve are those known as separable differential equations. To solve this type of differential equation, it helps to remember we can write the first derivative of y with respect to x as dy over dx. Now in general, a first-order differential equation can be written in the form dy over dx equals something. But now suppose dy over dx is equal to f of x times g of y, where here f is a function of x only, and g is a function of y only. And here the variables x and y can be separated, and so we say we have a separable differential equation. So for example, let's take a look at this and determine whether it's a separable differential equation. And so the thing we note is that we have dy over dx equals something, and our something can be factored. And note here, if we take f of x equals x and g of y equals y squared plus 1, we can write our differential equation in the form function of x times function of y. And so this is a separable differential equation. Or it could take a different differential equation. Now the first thing we might want to do is get this into standard form. In other words, we want to solve for the highest-order derivative, which is just the first derivative here. So let's rearrange things and solve for dy dx. And so the question we got to ask is, can we write our derivative as a product? So let's factor. And while we can factor this, there appears to be no way to write this as a product of a function of x with a function of y. And so this is not a separable differential equation. So why does this matter? Suppose we have a separable differential equation. In other words, we can write the derivative as a product of a function of x with a function of y. Now you should have been told in calculus that dy over dx is not a fraction. And so we can't break this up into a dy part and a dx part. Except you are probably also told it's sometimes helpful to view dy over dx as a fraction, where dy is our numerator and dx is our denominator. If we do that, we might consider taking the following steps. First of all, let's get rid of this denominator by multiplying both sides by dx. And then simplifying, dy over dx is not a fraction. So that legal disclaimer out of the way will pretend it's a fraction. And let's get our g of y term over here to the left hand side. And the thing to recognize here is that the reason that we call these separable differential equations is that we can rewrite them in a form where we have separated the variables. So over here on the left, we only have y and dy. And here on the right, we only have x and dx. Since each side only contains its differential variable, both sides can be anti-differentiated. So we'll anti-differentiate both sides, which allows us to solve the differential equation. So let's try to solve the differential equation dy over dx equals 5y. So the idea here is we want to separate the variables. All the x variables on one side, all the y variables on the other, and the constants, who cares? They'll end up on one side or the other, and we don't really worry about them. So from our differential equation, we'll multiply both sides by dx. And again, dy over dx is not a fraction, but it's sometimes helpful to view it as a fraction. We can simplify. And we have a y over on the right hand side, so we want to get that over onto the left. And so separating the variables gives us the equation 1 over y dy equals 5 dx. And so now we can anti-differentiate both sides. And at this point, it's useful to remember that the differential variable is controlling. So here on the left hand side, our differential variable is y. The only variable we're allowed to have on the left hand side is y, and we have it. On the right hand side, our differential variable is x, so the only variable we're allowed to have on the right hand side is x. And we have that as well. And so now we can proceed to anti-differentiate. So we'll find the anti-derivatives. And remember, we get constants of anti-differentiation every time we find an anti-derivative. And since our goal is to find y, let's go ahead and solve this equation for y. So we'll do a little bit of algebra. And we get y equals to some horrifying mess. However, we can simplify this a little bit further. Remember that c1 and c2 are constants of anti-differentiation, but they are constant. And so that means e to the power c2 is a constant. e to the power c1 is also a constant. And so their quotient, e to the power c2 over e to the power c1, well, that's just some constant, which we'll call c. This leads to an important idea when dealing with differential equations. Ordinarily, we must include a constant of anti-differentiation in any anti-derivative. But rather than apply the algebra to the constant all the way to the end, we can rely on the following principle. Any algebra or arithmetic done to a constant produces a constant. And what this means is that so long as we're not differentiating or anti-differentiating, all of our constants can be collapsed as we go. And in particular, when we're dealing with separable differential equations, the two constants we would ordinarily get from our initial anti-derivative can be combined into a single constant, which we traditionally put with the x function. So let's solve this differential equation. So let's separate the variables. We'll try to rewrite our equation, so all of our y terms are with the dy, and all of the x terms are with the dx. So again, a good first step here is we'll multiply both sides by dx. This y is with the dx, but we really want it to be with the dy, so we'll divide both sides by y. x squared is with the dy, but we want it to be with the dx, so we'll divide both sides by x squared. And simplify. So my variables are separated. Now I can anti-differentiate. So let's anti-differentiate both sides. So over on the left, we'll get log y, and there is a constant, but we'll let it be absorbed by the anti-derivative on the right. On the right we get... And let's solve this equation for y. Do a little algebra. And e to power c is some constant, which we'll just call c. And while we don't ordinarily like using the same variable between two different things, in some sense this c doesn't exist. It's just a placeholder that allows us to get to the general solution. y equals c times e to power minus 1 over x.