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anonymous
 one year ago
For question 2K4 (b) we're asked to find out all solutions for the two dimensional Laplace equation of the form w = ax^3+bx^2y + cxy^2 +dy^3. (Link
http://ocw.mit.edu/courses/mathematics/1802scmultivariablecalculusfall2010/2.partialderivatives/partafunctionsoftwovariablestangentapproximationandoptimization/problemset4/MIT18_02SC_SupProb2.pdf)
After equating w(xx) +w(yy) = 0, I get x(3a+c)+y(3d+b)=0. How do I get all solutions from here?
anonymous
 one year ago
For question 2K4 (b) we're asked to find out all solutions for the two dimensional Laplace equation of the form w = ax^3+bx^2y + cxy^2 +dy^3. (Link http://ocw.mit.edu/courses/mathematics/1802scmultivariablecalculusfall2010/2.partialderivatives/partafunctionsoftwovariablestangentapproximationandoptimization/problemset4/MIT18_02SC_SupProb2.pdf) After equating w(xx) +w(yy) = 0, I get x(3a+c)+y(3d+b)=0. How do I get all solutions from here?

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IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1you can now write \(W(x,y) = ax^3+bx^2y + cxy^2 +dy^3\) as: \(W(x,y) = a \ x^3  3d \ x^2y  3a \ xy^2 +d \ y^3 \), now knowing that this satisfies the Laplace Eqn, so: \(W(x,y) = a(x^3  3xy^2) + d(y^3  3x^2y)\) thus: \(W(x,y) = c_1(x^3  3xy^2) + c_2(y^3  3x^2y) = c_1.f_1(x,y) + c_2.f_2(x,y)\)
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