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

Worked example: separable equation with an implicit solution | Khan Academy


2m read
·Nov 11, 2024

We're given a differential equation right over here: cosine of y + 2, this whole thing times the derivative of y with respect to x is equal to 2x. We're given that for a particular solution, when x is equal to 1, y of 1 is equal to zero. We're asked, what is x when y is equal to π?

The first thing I like to look at when I see a differential equation is, is it separable? Can I get all the y's and dy on one side, and can I get all the x's and dx's on the other side? This one seems like it is. If I multiply both sides by dx, where you can view dx as the X differential of an infinitely small change in x, well then you get cosine of y + 2 * dy is equal to 2x * dx.

So just like that, I've been able to— all I did is I multiplied both sides of this times dx, but and I was able to separate the y's and the dy from the x's and the dx's. Now I can integrate both sides. So if I integrate both sides, what am I going to get?

The anti-derivative of cosine of y with respect to y is sine of y. Then the anti-derivative of two with respect to y is 2y. That is going to be equal to—well, the anti-derivative of 2x with respect to x is x^2. We can't forget that we could say a plus a different constant on either side, but it serves our purpose just to say plus C on one side.

So this is a general solution to this separable differential equation, and then we can find the particular one by substituting in when x is equal to 1, y is equal to 0. Let's do that to solve for C. So we get, or when y is equal to 0, x is equal to 1.

So sine of 0 + 2 * 0— all I did is I substituted in the zero for y— is equal to x^2. Well now, x is 1, so sine of 0 + C. Well, sine of 0 is 0, 2 * 0 is 0— all of that’s just going to be zero. So we get 0 is equal to 1 + C, or C is equal to -1.

So now we can write down the particular solution to this differential equation that meets these conditions. So we get, let me write it over here: sine of y + 2y is equal to x^2 - 1.

Now, what is x when y is equal to π? So sine of π + 2π is equal to x^2 - 1. Sine of π is equal to 0, and so we get—let's see, we can add one to both sides and we get 2π + 1 is equal to x^2.

Or we could say that x is equal to the plus or minus square root of 2π + 1. So I would write the plus or minus square root of 2π + 1, and we're done.

More Articles

View All
Introduction to spectroscopy | Intermolecular forces and properties | AP Chemistry | Khan Academy
In this video, we’re going to talk about spectroscopy, which is all about the interactions between light and matter. When we’re talking about light, we’re not just talking about visible light; we’re talking about electromagnetic radiation in general. So, …
Multiple points of influence due to separation of powers and checks and balances | Khan Academy
In several videos, we have touched on the idea of separation of powers between three branches of government in the United States. You have the legislative branch that writes laws and decides on the budget for the government. You have the executive branch …
Howard Marks: 50 Years of Investing Wisdom in 50 Minutes (Priceless Lecture)
Well, Cain said it best of anybody. He said, “Markets can remain irrational longer than you can remain solvent.” MH and some… so somebody who bets that a market which is irrational is going to… a market is too high, we say that’s irrational. Somebody who …
INSIDE a Spherical Mirror
Hey, Vsauce. Michael here. But you are actually right there. Well, at least the camera is. Mirrors are amazing. In fact, the word “mirror” comes from Latin “mirari,” meaning “to wonder at, to admire.” It’s also where we get the word miracle. Mirror- -acl…
Cave Diver vs. Tricky Maya Elves | Campfire Stories
I work in lots of sonatas in caves. The North day is basically a flooded cave, and I by myself, and I hear this lation. Whose it was there, and nobody answer? And then I heard a splash again, and I even have waves. I swear I have waves, the words: what’s …
Could Biking in a City Be Bad for Your Health? | National Geographic
Air pollution is bad for you, and we know that exercise is good for you. But there’s this unanswered question: is exercising in close proximity to traffic enough of a bad thing for you that we should be recommending separating biking lanes from traffic al…