, y) =g2(y).b) Of course, your answer in part a) is really, really ugly. But there is a nice, pretty answerforu(x, y). Can you find a really simple expression ofu(x, y)?9)(10 points) Consider the PDE∂2u∂x2+∂2u∂x∂y+∂2u∂y2= 0a) Verify that this PDE is elliptic.b) Perform a change of variables on this PDE:α=ax+byβ=cx+dyChoose the constantsa,b,c, anddin such a way to makeuαα+uββ= 0. In other words,this PDE is just a change of basis away from Laplace’s equation.c) The cubic polynomialu(x, y) =x3-3xy2is a solution to Laplace’s equation. Determinea cubic polynomial that is a solution touxx+uxy+uyy= 0.10(10 points) Eh, nothing is inspiring me. Ten free points for everyone!!!
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