(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

Listed above.

3. The attempt at a solution

first, I found [tex] u_{xx} [/tex] and [tex] u_{yy} [/tex]

[tex]u_{xx} = 2A [/tex]

[tex]u_{yy} = 2C [/tex]

From [tex] u_{xx} + u_{yy} = 0 [/tex] and the above results, I can get [tex] 2A + 2C = 0 [/tex].

Now, I plugged in the boundary condition:

[tex] cos(2\theta) + sin(2\theta) = Acos^2(\theta) + Bcos(\theta) sin(\theta) + Csin^2(\theta). [/tex]

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 [tex] x^2 + y^2 = 1[/tex] 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.

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# Homework Help: PDE's - Finding a certain solution to Laplace's equation on a circle

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