The dual problems of
utility maximization & expenditure minimization
Using the interactive graphs linked below, try changing any of the parameters to see the ordinary (Marshallian) and compensated (Hicksian) demand response.
Try increasing the price of x, and notice how both the ordinary and the compensated quantity demanded for x decrease, but x^* decreases faster than x^c (ordinary demand is more elastic).
Notice also that (when Px increases) the compensated response for y is positive, but the ordinary response is zero: this is because the income effect exactly cancels out the substitution effect, thus the cross-price elasticity of demand for Cobb-Douglas is zero: x and y are neither (gross) complements nor substitutes.
Because x enters non-linearly and y enters linearly in the utility function, MRS will only depend on x. This means that the ratio of prices is sufficient to determine the quantity of good x consumed. Then x^c=x^*, and the income effect on x is zero.