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

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


3m read
·Nov 11, 2024

What we're going to do in this video is see if we can solve the differential equation: the derivative of y with respect to x is equal to x times y. Pause this video and see if you can find a general solution here.

So, the first thing that my brain likes to do when I see a differential equation is to say, hey, is this separable? And when I say separable, can I get all the y's and dy's on one side and all the x's and dx's on the other side? You can indeed do that if we treat our differentials like if we could treat them like algebraic variables, which is fair game when you're dealing with differential equations like this.

We could multiply both sides by dx. So, multiply both sides by dx and we could divide both sides by y. Let me move this over a little bit so we have some space. So, we could also multiply both sides by 1 over y, 1 over y. And what that does is these dx's cancel out, and this y and 1 over y cancels out.

We are left with, let me write all the things in terms of y on the left-hand side in blue. So, we have 1 over y dy is equal to, and then I'll do all this stuff in orange. We have: is equal to... we're just left with an x and a dx, x dx. And then we’ll want to take the indefinite integral of both sides.

Now, what's the antiderivative of 1 over y? Well, if we want it in the most general form, this would be the natural log of the absolute value of y, and then this is going to be equal to the antiderivative of x, which is x squared over 2. And then we want to put a constant on either side; I'll just put it on the right-hand side plus c. This ensures that we're dealing with the general solution.

Now, if we want to solve explicitly for y, we could raise e to both sides power. Another way to think about it is if this is equal to that, then e to this power is going to be the same thing as e to that power.

Now, what does that do for us? Well, what is e to the natural log of the absolute value of y? Well, I'm raising e to the power that I would have to raise e2 to get to the absolute value of y. So, the left-hand side here simplifies to the absolute value of y, and we get that as being equal to...

Now, we could use our exponent properties. This over here is the same thing as e to the x squared over 2 times e to the c. I am just using our exponent properties here. Well, e to the c we could just view that as some other type of constant; this is just some constant c.

So, we could rewrite this whole thing as c e e to the x squared over 2. Hopefully, you see what I'm doing there. I just use my exponent properties: e to the sum of two things is equal to e to the first thing times e to the second thing.

And I just said, well, e to the power of some constant c we could just relabel that as, let's call that our blue c. And so, this simplifies to blue c times e to the x squared over 2.

Now, we still have this absolute value sign here, so this essentially means that y could be equal to... We could write it this way: y could be equal to plus or minus c e e to the x squared over 2.

But once again we don't know what this constant is; I didn't say that it was positive or negative. So, when you say a plus or minus of a constant here, you could really just subsume all of this. I'll just call this with red c, so we could say that y is equal to... I’ll just rewrite it over again for fun: y is equal to red c, not the red c, but a red z times e to the x squared over 2.

This right over here is the general solution to the original separable differential equation.

More Articles

View All
How To Not Give A F*** | Stoic Exercises For Inner Peace
I’m not really into swearing on my channel, but I think that there’s no better contemporary expression for being indifferent towards what people think about you than the words: I don’t give a f***. The problem is that not giving a f*** isn’t always a goo…
The 7 BEST Side Hustles That Make $100 Per Day
What’s up, Graham? It’s guys here, and welcome to the most requested topic I get here in the channel, and that would be side hustles. I get it; it’s easy to see the appeal of making money in your spare time. After all, chances are most of us have an extra…
Cosine equation algebraic solution set
The goal of this video is to find the solution set for the following equation: negative 6 times the cosine of 8x plus 4 is equal to 5. And like always, I encourage you to pause this video and see if you can have a go at this before we do it together. A re…
Visually dividing decimal by whole number
In this video, we’re going to try to figure out what 4 tenths divided by 5 is. So pause this video and see if you can think about it before we work through it together. We’re really going to think about approaching this visually. All right, now let’s wor…
The Challenges with Cancer Trials | Breakthrough
ANDRE CHOULIKA: We didn’t have any intention of injecting these type of vials to patient because we needed a lot of vials to be able to file our clinical trial application. And this was planned to be done with the University College London. NARRATOR: Bef…
Where Does The Sun Get Its Energy?
It’s another beautiful day in Sydney. I’m out enjoying the sun and looking to find out if anyone knows why the sun shines? So the sun gives us a lot of… ? Heat? Sunshine? Provides energy? (yes) Heat energy. Where does the sun get that energy from? I…