yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Worked example: separable differential equation (with taking log of both sides) | Khan Academy


2m read
·Nov 11, 2024

Let's say we need to find a solution to the differential equation that the derivative of y with respect to x is equal to x squared over e to the y. Pause this video and see if you can have a go at it. I will give you a clue: it is a separable differential equation.

All right, now let's do this together. So whenever you see any differential equation, the first thing you should try to see is: is it separable? When I say separable, I mean I can get all the expressions that deal with y on the same side as the dy, and I can separate those from the expressions that deal with x, and they need to be on the same side as my differential dx.

So how can we do that? Well, if we multiply both sides by e to the y, then e to the y will go away right over here, so we will get rid of this y expression from the right-hand side. Then we can multiply both sides by dx.

So if we did that, let me move my screen over a bit to the left. I’m going to multiply both sides by e to the y, and I’m also going to multiply both sides by dx. Multiplying by dx gets rid of the dx on the left-hand side, and it sits on the right-hand side with the x squared. So all of this is now e to the y dy is equal to x squared dx.

Just the fact that we were able to do this shows that it is separable. Now what we can do now is integrate both sides of this equation.

So let's do that. What is the integral of e to the y dy? Well, one of the amazing things about the expression, or you could say the function, something is equal to e to the — and normally we say e to the x, but in this case it’s e to the y — is that the anti-derivative of this is just e to the y. We’ve learned that in multiple videos; I always express my fascination with it.

So this is just e to the y, and likewise, if you took the derivative of e to the y with respect to y, it would be e to the y. Remember this works because we are integrating with respect to y here. So the integral of e to the y with respect to y is e to the y, and that is going to be equal to the anti-derivative of x squared.

Well, that is, we increment the exponent, so that gets to x to the third power, and we divide by that incremented exponent. Since I took the indefinite integral of both sides, I have to put a constant on at least one of these sides. So let me throw it over here, plus c.

Just to finish up, especially on a lot of examinations like the AP exam, they might want you to write in a form where y is explicitly an explicit function of x. So to do that, we can take the natural log of both sides.

So we take the natural log of that side, and we take the natural log of that side. Well, the natural log of e to the y — what power do we have to raise e to get to e to the y? Well, that's why we took the natural log; this just simplifies as y.

And we get y is equal to the natural log of what we have right over here: x to the third over three plus c. And we are done.

More Articles

View All
Harris Proposes $50k Tax Break For Small Businesses
You’ve helped entrepreneurs jump start their small business. There’s also this proposal about a $50,000 tax deduction for businesses. How does that sound to you? Look, I’m very happy that you talked about small business because you got to remember her ad…
Warren Buffett: How to Make Money During Inflation
Are you seeing signs of inflation beginning to increase? We’re seeing very substantial inflation. It’s very interesting. I mean, we’re raising prices, people are raising prices to us; it’s being accepted. I mean, inflation is a big concern for everyone ri…
Kinetic molecular theory and the gas laws | AP Chemistry | Khan Academy
In other videos, we touched on the notion of kinetic molecular theory, which I’ll just shorten as KMT. It’s just this idea that if you imagine a container—I’ll just draw it in two dimensions here—that it contains some gas. You can imagine the gas as being…
Khan Academy Live: SAT Math
Hello and welcome to Khan Academy live SAT. I’m Eric, I’m an SAT tutor and one of the SAT experts here at Khan Academy, and I’m so excited to be with you today and over the course of the next few weeks as we cover SAT Math, reading, and writing with one c…
Languages Are Dying: Here’s Why We Should Be Worried
First words, they’re special regardless of the language you speak or the sounds that you’re able to make. First words can be many different things. For parents, they can be the realization of a dream or the start of a new chapter for the next great pionee…
6 Millionaire Habits I Wish I Knew At 20
What’s up you guys, it’s Graham here. So I know a lot of people say your 20s are the most transformative and influential years of your entire life, and I have to say it, but that is absolutely a load of truth. Because looking back over my last 10 years, I…