If x(y,t) is a function of y and t and derivatives of x with respect to by y and t exist and are continuous, then the order of differentiation doesn't matter. This is called Clairaut's Theorem. I am sure I am missing some of the conditions and requirements, but in your case we would have d^(2)x/(dydt) = d^(2)x/(dtdy). This comes from multivariable calculus, and those "d"s should be "∂"s (partial derivatives), but hopefully that is good enough for your purposes.