Find a solution of Laplace's equation $\displaystyle u_{xx} + u_{yy} = 0 $ of the form $\displaystyle u(x,y) = Ax^2 + Bxy + Cy^2 \ (A^2 + B^2 + C^2 \not= 0 ) $ which satisfies the boundary condition $\displaystyle u(cos(\theta),sin(\theta)) = cos(2\theta) + sin(2\theta) $ for all points $\displaystyle (cos(\theta),sin(\theta)) $ on the circle, $\displaystyle x^2 + y^2 = 1 $.

first, I found $\displaystyle u_{xx} $ and $\displaystyle u_{yy} $

$\displaystyle u_{xx} = 2A $

$\displaystyle u_{yy} = 2C $

From $\displaystyle u_{xx} + u_{yy} = 0 $ and the above results, I can get $\displaystyle 2A + 2C = 0 $.

Now, I plugged in the boundary condition:

$\displaystyle cos(2\theta) + sin(2\theta) = Acos^2(\theta) + Bcos(\theta) sin(\theta) + Csin^2(\theta). $

I tried various trig substitutions here and couldn't seem to get anywhere. However, with this equation and the one above, I have two equations (but there are three unknowns). I am pretty sure I have to use the $\displaystyle x^2 + y^2 = 1$ to write another equation so that I can solve for A, B, and C, but I do not know how to use the circle information.

Any help would be greatly appreciated.

Thanks in advance.