I am trying to solve: $$ r^2 \frac{ \partial^2 u}{\partial r^2} + r \frac{\partial u}{ \partial r} + \frac{ \partial^2 u}{\partial \theta ^2} = 0 $$ I have a chosen a separable anszats of the form $u(r, \theta ) = R(r) \Theta(\theta)$ and have arrived to this:

$$ \frac {r^2} R \frac {d^2R}{dr^2} + \frac rR \frac{dR}{dr} = -\frac 1 \Theta \frac{d^2\Theta}{d\Theta^2} =\lambda$$

My goal is to prove that $$u(r, \theta) = \sum_{n=0}^{\infty} (A_n r^n + B_n r^{-n})(C_n \cos (n \theta) + D_n \sin(n\theta))$$

I know will be able to solve the radial equation using a trial of the form $r^{\alpha} $ but what confuses me is the constant $\lambda$ I don't understand why it has to be an integer, and why, for example it can't be a complex number. How do I know that $ \lambda$ has to be such that the angular part is periodic and oscillating? These constants are what really confuse me in PDE's overall.

  • 1
    $\begingroup$ Note that $\theta$ and $\theta+ 2n\pi$ represent the same point in the $x, y$ plane. So we must have $u(r, \theta) = u(r, \theta + 2n\pi)$ (in order that $u$ is really a function defined in the $x-y$ plane. . $\endgroup$ Oct 9 '20 at 15:34
  • $\begingroup$ What region are you trying to solve this on? $\endgroup$ Oct 12 '20 at 5:09

As mentioned some solutions are
$$u(r, \theta) = \sum_{n=0}^{\infty} (A_n r^n + B_n r^{-n})(C_n \cos (n \theta) + D_n \sin(n\theta)) \tag 1$$ But that is far to all solutions. The index $n$ isn't necessarily integer, say : $$u(r, \theta) = \sum_{\forall \nu} (A_\nu r^\nu + B_\nu r^{-\nu})(C_\nu \cos (\nu \theta) + D_\nu \sin(\nu\theta))$$ Or even more general : $$u(r, \theta) = \int (A(\nu) r^\nu + B(\nu) r^{-\nu})(C(\nu) \cos (\nu \theta) + D(\nu) \sin(\nu\theta)) d\nu$$ with arbiytrary real and/or complex functions $A(\nu),B(\nu),C(\nu),D(\nu)$.

If the solutions are only on the form $(1)$ this is because some context and/or some implicit conditions. Also the solutions are not necessarily oscillating. Again if they are oscillating this is because some context and/or some implicit conditions. One cannot definitively answer without the full description of the problem.


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