Separable Differential Equations - Example 2

Here is a graph depicting a family of ellipses centered about the origin.

Created with: www.desmos.com/calculator

In this tutorial, we will solve the following first order, non-linear differential equation...

...and see how the result relates to be above graph.

The first thing to note is this is a separable differential equation because we can get all of the x's to the right-hand side, and leave all of the y's on the left. Thus, rearranging, we have...

We can write this in differential form by getting the dx to the right as well...

Now, we can integrate both sides...

Ok, we can now "pretty" this result up a bit...

Now what the implicit solution...

...describes is a family of ellipses centered about the origin, with aspect ratio 5:3. We've graphed the ellipses with varying values for R²

That will complete this tutorial. Below is a list of tutorials I've created so far on Differential Equations:

1. Introduction to Differential Equations - Part 1

2. Differential Equations: Order and Linearity

3. First-Order Differential Equations with Separable Variables - Example 1

4. Separable Differential Equations - Example 2

5. Modelling Exponential Growth of Bacteria with dy/dx = ky

6. Modelling the Decay of Nuclear Medicine with dy/dx = -ky

7. Exponential Decay: The mathematics behind your Camping Torch with dy/dx = -ky

Please give me an Upvote and Resteem if you have found this tutorial helpful.

Please ask me a maths question by commenting below and I will try to help you in future videos.

I would really appreciate any small donation which will help me to help more math students of the world.

Tip me some DogeCoin: A4f3URZSWDoJCkWhVttbR3RjGHRSuLpaP3
Tip me at PayPal: https://paypal.me/MasterWu