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anonymous
 one year ago
For qn 2K3 (b) we are supposed to find all solutions of the two dimensional Laplace equation of the form w = ax^3 + bx^2y + cxy^2 + dy^3.
Equating w(xx) + w(yy) = 0, I get x(3a + c) + y(3d + b) = 0. How do I find all solutions from here? Link to the question
http://ocw.mit.edu/courses/mathematics/1802scmultivariablecalculusfall2010/2.partialderivatives/partafunctionsoftwovariablestangentapproximationandoptimization/problemset4/MIT18_02SC_SupProb2.pdf
anonymous
 one year ago
For qn 2K3 (b) we are supposed to find all solutions of the two dimensional Laplace equation of the form w = ax^3 + bx^2y + cxy^2 + dy^3. Equating w(xx) + w(yy) = 0, I get x(3a + c) + y(3d + b) = 0. How do I find all solutions from here? Link to the question http://ocw.mit.edu/courses/mathematics/1802scmultivariablecalculusfall2010/2.partialderivatives/partafunctionsoftwovariablestangentapproximationandoptimization/problemset4/MIT18_02SC_SupProb2.pdf

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IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1have replied to this on another forum subheading http://openstudy.com/study#/updates/5597d85be4b0105c3c3323de in short, you now know that \(c = 3a\) and \(b = 3d\) ensures the polynomial satisfies the Laplace Eqn \(\nabla^2 = 0\), so if build in the fact that \(c = 3a\) and \(b = 3d\), then you have a solution with 2 constants, \(c_1\) & \(c_2\) and 2 distinct functions \(f_1(x,y)\) and \(f_2(x,y)\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Thanks! It makes sense now. I don't know why but I didn't equate the coefficients of x and y to be 0 when equating ∇^2=0. Thanks a lot again!
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