# Change of variables of differential equation

1. May 23, 2012

### climbon

I have an equation that I am trying to change the variables of, it has the form;

$$\frac{d}{dt} W = g \frac{\partial}{\partial y} U + h x \frac{\partial^2}{\partial y^2} Z$$

Where W, U and Z are my dependent variables (This equation is just one of 3 coupled equations but have written only the above in general form).

I want to change the variables into a rotating frame, so;

x -----> x' = x(0) cos(wt) + y(0) sin(wt)
y -----> y' = -x(0) sin(wt) + y(0) cos(wt)

When putting these into the top equation, obviously it is simple to replace the x and y's but how do I change the differentials w.r.t. y with the new variable?

Thanks for any help :D

2. May 23, 2012

### Vargo

So you have a change of variables that looks like:
x'=x'(x,y,t)
y'=y'(x,y,t)
t'=t

Chain rule:

df/dy = df/dx' * dx'/dy + df/dy'*dy'/dy + df/dt'*dt'/dy= sin(wt)df/dx' +cos(wt)df/dy'

Sorry, I'm not sure how to use latex here. d is supposed to mean a partial d.