• anonymous
For qn 2K-3 (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-
OCW Scholar - Multivariable Calculus
  • Stacey Warren - Expert
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  • chestercat
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  • IrishBoy123
have replied to this on another forum subheading 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
Thanks! 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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