9.3 Separable Equations

Hello friends. I was thinking that we'd talk about Separable Equations today. Doesn't that sound nice? I think so.

Separable equations are differential equations of the first order (i.e. equations that contain a function and its derivative) that we can solve explicitly by separating (wow) the two variables involved into the two sides of the equation and by using integrals on both sides to get rid of the dx's and dy's. (Brainstorm: what if there were a three-sided equals sign? Wouldn't that be cool? Or would it just be pointless?)

There are certain steps we need to follow in order to get full credit on the AP's Free Response Questions. Don't worry, I'll cover them. How about an example? I think so.

Find the solution of the differential equation that satisfies the given condition.

(dy/dx) = y² + 1, where y(1) = 0.

Step One: Separate the variables using algebra.
(dy/ (y² + 1)) = dx

Steps Two and Three: Take the antiderivative of both sides. (⌠= integral sign)
⌠(dy/ (y² + 1)) = ⌠(1dx)
tan^-1y = x

NOTE - The integral of the left side of the equation is one of the rare occurrences of a derivative of an inverse trigonometric function. Yes, it's annoying. (sad face).

Step Four: Recognize the constant of integration.
tan^-1y = x + C

Step Five: Solve for "C" using the given condition (y(1) = 0).
tan^-1(0) = (1) + C
0 = 1 + C
C = -1
Good job.

Step Six: We end up with tan^-1y = x - 1. Our book seems to like solving for y at this point. I'm not sure if this is really necessary, but being the brilliant maths students we are, why not do it?

y = tan (x - 1). This is the answer you've all been waiting for, or, rather, for which you've all been waiting.

SOPHIE. Wake up and take notes; you're posting next.

And, if you couldn't handle my colloquial math tongue, this website presents separable equations in more textbooky way. It also has some sample problems you can try, if by chance you've done all of the ones in our own textbook.

Here is a video that my brother made for his Chinese class at Vassar College. It's about drugs and their effect on the psyche. Not really, but it is about drugs. And it's in Chinese, so it's funny. And it has a cute old Chinese man at the end with a cool voice. I hope you enjoy it, even if their accents sound like an electronic dictionary (Laurie).

More Drugs, More Chinese Problems

Bye bye.