 In this video, we will discuss the solution to question 12 from the practice midterm exam, which asks us to consider the separable differential equation dy over dx equals e to the x over e to the y. And this question comes in two parts. First, we're told to demonstrate how to separate the variables in the above equation. And that includes the differentials that are part of it, right? So we're trying to separate the variables here. If this is a separable differential equation, we should separate the variables. In this situation, we basically just get away with cross multiplication, right? Because we want it to look something like the following. Our goal is to make it look like dy over g of y is equal to, I guess I take it back. We want this to look like g of y times dy is equal to f of x dx. And it could be that the function is in the denominator. That's perfectly fine here. And so for this one, we really just have to cross multiply. That does it for us. And so we see that dy over, well, I guess not over. We're going to get dy times e to the y is equal to e to the x dx. And so not a lot to show there, but that's what we get for separation of variables. e to the y dy is equal to e to the x dx. Now admittedly, if you saw an alternate version of this question on your final exam, the separation of variables process could be much more complicated than this. The point is to get all of the y's on the left side and all the x's on the right hand side. And so then that gives us the part A, and that's four points right there. The next four points then comes from solving the differential equation using these, using the separation of variables. So we want to integrate e to the y dy is equal to integral of e to the x dx. So integrate both sides of the equation. This gives us, well, not so bad, we're going to integrate the left hand side with respect to y. So we get e to the y on the right hand side, we're going to get e to the x. You do need to have a constant. You do not have to write the constant on both sides. It's sufficient just to take the constant on the right hand side because our goal is to solve for y. You do want to solve for y in this situation. For which case to get rid of the y, we need to take the natural log on both sides. And so we end up with y on the left hand side. We're going to get the natural log of e to the x plus a constant like so. And that's really, that's really all we can do. I mean, there's nothing more we can simplify. I mean, we're very tempted because we have an e to the x inside of the natural log. You might be very tempted to cancel that out and say something like y equals x plus c. But that is totally bonkers. That's not the right answer whatsoever. And so this actually kind of paints the picture why it's important to include that plus c very early on. We sometimes get in the bad habit when we work with anti-derivatives, it's like, oh, I calculate anti-derivative. I'm good to go. It took me a while to do it. Oh, wait. I forgot the plus c. You just throw in the plus c at the very end. You're good. In this situation, if you forgot to put the plus c in here, you'd be very tempted to say y equals x because that's how it's simplified. And then you're like, oh, wait, I need to do a plus c, right? And so if you say y equals x plus c, that would be incorrect. The correct answer would be y equals the natural log of e to the x plus a constant. And so that solved this one. This one turned out to be a very simple, a several differential equation. Don't try to measure the simplicity of this, the practice questions to be any measurement of the quality of the difficulty on the actual test. The ones on the practice test could be crazy hard. The ones on the practice test could be crazy simple. Or they could be somewhere in the middle, right? And this is because on the actual test, some questions will be easier, some will be harder. Not every question is going to be super hard, at least it's not supposed to be. Not every question should be super easy as well. There'll be a balance. And so do anticipate, you might need to practice some harder versions of this question as you prepare for the final exam.