If the going gets rough when doing a math exercise, it's always worth checking whether you copied the question correctly. I'm wondering whether those fourth powers should really be squares. That would make things a lot less unpleasant.
Assuming that they really are fourth powers, you just have to roll up your sleeves and persevere with the calculations. Your partial derivative equations tell you that , and . Then the equation becomes , from which . Then , and x and y are the same expression multiplied by .
So it's all a bit messy, but not impossibly hard.