Need help here with figuring out boundary conditions for this problem. Also, for (i), I do know a general way or method but here I am confused since from both equations how do I find out my desired result?
Also, it says $u(x) = v(r)w(*)$, (*----I dont know what's this and which syntax you use to define it on latex).
this equation doesnt make sense to me since it should be such that you can take 1st and 2nd partial derivatives of it but here how do I take it? Maybe its $x$ is a function of $r$ and the other variable but its not mentioned in the problem.
I came up with a solution which I think looks correct but it doesn't match with whatever is expected in (i).
It would be great if someone helps me on this. Appreciate your help & support.
The construction $u(x)=v(r)w(\theta)$ is an ansatz, an educated guess. There's no a priori reason for it, and depending on the boundary conditions the solution to the equation might not have this form.
However, it is very useful as a building block for the general solution to the problem. The variable $x$ is simply a short-hand for the pair $r$ and $\theta$.
Hint: Use the ansatz to express the partial derivatives as total derivatives of $v$ and $w$, then put everything that depends on $r$ on the LHS, and everything that depends on $\theta$ on the RHS. What can be said about the value these functions are equal to?