I have to solve this boundary value problem for
$$ y''(x) -x \ y(x) = 0 \\ y(a) = y_a \\ y(b)=y_b $$
for some $y_a$ and $y_b$. I have to use the Shooting method with Runge–Kutta 4.
So far I have
$$ z_1(x)=y(x) \\ z_2(x)=y'(x) $$ from which I get: $$ z_1'(x)=z_2(x) \\ z_2'(x)=x\ z_1(x) $$
And this is where it stops. What am I missing to have a full system of equations that I'll solve with Rk4? And how do I get the parameter $\alpha$ that I will have to tune?
Sorry if the question isn't well formed, but I am still quite lost here. I have looked at some other questions about shooting method but the answers just confuse me even more (it may be just because of the late hour), so a simple answer or a hint would be nice.