The problem is as follows:
Solve Laplace's equation $$u_{xx} + u_{yy} = 0$$ for a rectangular plate subject to the given boundary conditions: $$u(0,y) = 0, \ u(a,y) = 0, \ \frac{\partial u}{\partial y} \bigg\rvert_{y = 0} = 0 , \ u(x,b) = f(x)$$
This is somewhat of a simple problem to solve, however my questions arises when we begin to apply our boundary conditions to our Sturm-Liouville equations
\begin{align} X''(x) + \lambda^2X &= 0, \quad X(0) = X(a) = 0 \\ \implies X_{n}(x) &= A_{n} \sin(\lambda_{n} x) \end{align}
where $\lambda_n={{n\pi}\over a}$.
Why on earth don't we treat the $\left.{\partial u\over \partial y}\right\rvert_{y=0} =0$ as a boundary condition for $Y''(y) - \lambda^{2} Y(y) = 0$? My professor told me this is an incorrect approach, but using it as such gives us
\begin{align} Y(y) &= B \cosh(\lambda y) + C \sinh(\lambda y) \\ \implies Y'(y) &= \lambda B \sinh(\lambda y) + \lambda C \cosh(\lambda y) \\ \implies Y'(0) &= \lambda C \\ &= 0 \\ \implies C &= 0 \\ \implies Y_{n}(y) &= B_n \cosh \left( \frac{n \pi y}{a} \right) \\ \implies u(x,y) &= X_n(x) Y_n(y) \\ &= \sum_{n=1}^{\infty} B_n \cosh \left( \frac{n \pi y}{a} \right) \sin \left( \frac{n \pi x}{a} \right) \end{align}
Whereas when we apply it as an "initial condition" (?) the following happens:
\begin{align} u(x,y) &= X_n(x) Y_n(y) \\ &= \sum_{n=1}^{\infty} A_n \cosh \left( \frac{n \pi y}{a} \right) \sin \left( \frac{n \pi x}{a} \right) + B_n \sinh \left( \frac{n \pi y}{a} \right) \sin \left( \frac{n \pi x}{a} \right) \\ \implies u_y(x,0) &= \sum_{n=1}^{\infty} \left( \frac{n \pi}{a} \right) B_n \sin \left( \frac{n \pi x}{a} \right) \\ &= 0 \\ \implies B_n &= 0 \\ \implies u(x,y) &= \sum_{n=1}^{\infty} A_n \cosh \left( \frac{n \pi y}{a} \right) \sin \left( \frac{n \pi x}{a} \right) \end{align}
Which is the exact same as if we had used $\left.{\partial u\over \partial y}\right\rvert_{y=0} =0$ as a boundary condition! I was told by my professor that "It's a different problem!" and "You can only have a Sturm-Liouville problem in one dimension," but I'm still so confused since both approaches resulted in the same answer. (The constant coefficients are different of course, but that shouldn't matter when you begin applying Fourier Series to solve for the initial condition $u(x,b) = f(x)$). Also, I apologize if there are any formatting issues/small mistakes. This is my first time posting to Math Stack Exchange!
EDIT: Fixed an error regarding the initial condition in my conclusion paragraph