1. [SOLVED] Swapping your dependent/independent variables? (ODE)

I just discovered this board and it seems like an awesome idea. I have a question that I've been wracking my brains on for a while now and I can't find any answers from any other online sources so I was hoping that asking real people would help =).

I have a question set that goes like this:

Solve the following equations by regarding y as the independent variable rather than x

and an example of one of the questions is:

(1-4xy^2)dy/dx = y^3.

Now, how do you swap the dependent/independent variables?

Is the differential equation the same as

(1-4xy^2) = y^3(dx/dy)?

i.e., is dy/dx*dx/dy = 1?

Thanks.

2. Originally Posted by Friedeggs123
I just discovered this board and it seems like an awesome idea. I have a question that I've been wracking my brains on for a while now and I can't find any answers from any other online sources so I was hoping that asking real people would help =).

I have a question set that goes like this:

Solve the following equations by regarding y as the independent variable rather than x

and an example of one of the questions is:

(1-4xy^2)dy/dx = y^3.

Now, how do you swap the dependent/independent variables?

Is the differential equation the same as

(1-4xy^2) = y^3(dx/dy)? Mr F says: Yes.

i.e., is dy/dx*dx/dy = 1?

Thanks.
So after some further re-arranging you have

$\frac{dx}{dy} +\frac{4}{y} x = \frac{1}{y^3}$.

Now use the integrating factor method to solve for x = x(y).