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- 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

Question

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- 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

irishboy123 · Answer

have 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 · Answer

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!