 Welcome to this quick recap of section 7.4 on separable differential equations. A separable differential equation is a differential equation that can be put into this form. dy dt is equal to a function involving only y times a function involving only t. Remember that notation like this right here means that we have a formula using only the variable indicated only y or only t. Here's some examples of separable differential equations. We have this one in which the functions g and h are written exactly like they appear in the definition. But sometimes we have to do a little bit more work. Here's a differential equation that does not look like two functions multiplied together. However, if we do a little factoring, we can see that there's a g of y right here and an h of t right here. Finally, a formula like this is also a separable differential equation because we have a function of y, and then the function of t, it can be simply 1. 1 is after all a function of t. So why do we care about separable differential equations? The big idea here is that we can use separation of variables, a method based on separable differential equations, in order to analytically solve many common types of differential equations. To analytically solve, we mean to actually find exact solutions rather than estimates as we found with Euler's method. Here's the method for finding the solution to a separable differential equation. This method is called separation of variables. We'll present it here in a general form, and in the other videos you can see examples of how to use it. So to solve a separable differential equation, we begin with a separable differential equation, of course. Our next step is to divide both sides by g of y, so that we have all of the y terms on the left, and all of the t terms on the right. This is called separating the y's from the t's. Next, we'll integrate both sides of this new equation with respect to the variable t. Notice that we've added integral signs and dt's to both sides. At this point, we can recognize that we can simplify a little bit. We can convert this y term to a dy integral. This is actually a result of a substitution. Because we had a dy dt here, we're able to make a substitution to get rid of that derivative and turn our variable into a dy variable instead of a dt variable. This gives us two traditional-looking integrals, one with respect to y on the left, and one with respect to t on the right. And so our final step is to evaluate the antiderivatives and then solve for y. When we solve for y, we will have an analytic solution, that is, a formula that tells us exactly what y can be equal to for this differential equation. This will be our first way to actually get an exact solution. Along the way, we have to be careful with our plus c's. Separation of variables finds infinite families of solutions to differential equations, and so there will be some unknown constants. The antiderivative sign means that two functions are in the same family of antiderivatives, and so we only need one constant. In the example written here, which is a very typical example of these sort of antiderivatives you might find yourself doing with separation of variables, in this example, once we find the antiderivatives on the second line, we only need to put one plus c, because we know that the two antiderivatives differ from each other by a constant. There's no need to put a constant on both sides of this. Similarly, any expression involving only constants can be collapsed into a single constant. For example, you might end up exponentiating in order to solve a separable differential equation, and so you might end up with something like this e to the a right here, where a is an unknown constant. Well, e is a number, and a is a number, and so e to the a can be collapsed into a single constant, and by convention we usually do this to simplify our formulas. So we here we've replaced this e to the a with a single unknown constant c. It would also be a good idea to write something like c equals e to the a to make it easier to understand what you've done when simplifying. Notice, however, that the e to the t plus 2 cannot be simplified immediately, because t is a variable, and so we can't reduce it down as if it were a constant. Anything with a variable involved needs to be left in its fully expanded form. Now that we've seen this, let's take a look at how to actually solve some differential equations.