Derivation of a Demand Equation
hint:

Solve for optimal $x^*,y^*$ then substitute the solutions into budget constraint and then solve the budget constraint for $x^*,y^*$ respectively and you will get the demands as a function of the parameters and of course conditional on the optimal consumption of the other variable.

Knowing that we have an interior solution helps because that means we know demand(s) is(are) not zero.