You're missing a 2 in your answer. As for showing it satisfies the PDE you can either (1) try and solve for explicitly, then sub into the PDE or (2) differentiate your solution implicitly and solve for and and sub into your PDE.
I managed to get u(x,y) explicitly and got u_x and u_y.
In terms of your method (1), you say solve for u explicitly and then sub into PDE, but we have to get u_x and u_y beore we sub in right?
For your (2), this looks interesting but wouldnt it be difficult to get u_x and u_y if its implicitly defined? otherwise (1) is the exact same (2)?
Thanks