Y g 2 y b of course your answer in part a is really

, y ) = g
2
( y ).
b) Of course, your answer in part a) is really, really ugly. But there is a nice, pretty answer
for u ( x, y ). Can you find a really simple expression of u ( x, y )?
9) (10 points) Consider the PDE
∂ 2 u
∂x
2
+
∂ 2 u
∂x∂y
+
∂ 2 u
∂y
2
= 0
a) Verify that this PDE is elliptic.
b) Perform a change of variables on this PDE:
α = ax + by
β = cx + dy
Choose the constants a , b , c , and d in such a way to make u
αα
+ u
ββ
= 0. In other words,
this PDE is just a change of basis away from Laplace’s equation.
c) The cubic polynomial u ( x, y ) = x 3 - 3 xy 2 is a solution to Laplace’s equation. Determine
a cubic polynomial that is a solution to u
xx
+ u
xy
+ u
yy
= 0.
10 (10 points) Eh, nothing is inspiring me. Ten free points for everyone!!!