Characteristic Value Problem - Solutions to Characteristic Values

Question

I am given the characteristic value problem:

$\dfrac{d^2y}{dx^2}+2ky = 0$ , $y(0) = y(3) = 0$

And asked the following two questions. May anyone assist me in finding a way to answer these?

Using the method of finite differences:

1. What is approximately the value of $k$ with least magnitude for which the boundary value problem has non-zero solutions?
2. What are the approximate non-zero solutions corresponding to the characteristic values answered in 1?

I am very close the answering these two questions because I've already modeled the situation as follows:

If I apply the method of finite differences to this problem, with $\Delta x = 1$, I get the linear equation:

$ \left( \begin{array}{cc} 2k-2 & 1 \\ 1 & 2k-2 \end{array} \right) % \left( \begin{array}{cc} y_1 \\ y_2 \end{array} \right) = \left( \begin{array}{cc} 0 \\ 0 \end{array} \right) $

Where $y_1$ and $y_2$ denote the approximate values of $y$ at $x=1$ and $x=2$. The matrix

$A = \begin{pmatrix} -2 & 1\\ 1 & -2 \end{pmatrix}$

has eigen values $\lambda_1 = -1$ and $\lambda_2 = -3$

I know that I must use this in order to answer the two questions, so I was hoping the answer can proceed off the back of what I have calculated above.