Question

For question 2K-4 (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/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-a-functions-of-two-variables-tangent-approximation-and-optimization/problem-set-4/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?

Answer 1

you 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:
\(W(x,y) = c_1(x^3 - 3xy^2) + c_2(y^3 - 3x^2y) = c_1.f_1(x,y) + c_2.f_2(x,y)\)