Question

how do you handle an ODE that has both variables (x and y) inside a cosine? (i.e. dx/dy=1+cos^2(y-x))

Answer 1

If by ODE, you mean, "relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable." (Wikipedia: http://en.wikipedia.org/wiki/Ordinary_differential_equation), then you just do implicit differentiation.

http://tutorial.math.lamar.edu/Classes/CalcI/ImplicitDIff.aspx