Asymmetric (in sign) cross-price derivatives in consumer-theory problem
MWG proposition 3.G.2 is the one you’re thinking of, but that applies to Hicksian demand’s hessian (constant utility). What you solved for is the Walrasian demand, so we want to look at the Slutsky matrix.
$$S_{lk}(p,w) = \partial x_l(p,w) / \partial p_k + \partial x_l(p,w)/ \partial w * x_k(p,w)$$
First term: uncompensated change in demand of good l due to a change in $p_k$ (when multiplied by $\partial p_k$)
Second term: effect of wealth change on demand of good l (when
multiplied by $\partial p_k$) [https://scholar.harvard.edu/files/basilico/files/mwg_flashcards.pdf]
If you check the symmetry of this matrix (in our case l and k are x and y), I believe it should work out.