Change of Variables on a boundary differential equation

I have the following question on numerical analysis, but it's more related to ODE.

Using a change in variables, Show that the boundary value problem: $$-\frac{d}{dx}(p(x)y') + q(x)y = f(x), ~ a \leq x\leq b, ~ y(a) = \alpha , ~ y(b) =\beta$$ Can be transformed to the form: $$-\frac{d}{dw}(p(w)z') + q(w)z = F(w), ~ 0\leq w \leq 1, ~ z(0)=0 , ~ z(1)=0$$

I've had quite a headache with this problem. I've tried using $z(x) = y(x(b-a)+a)$, but I had trouble with the differentiation of $p(x)z'$.
I could use some help on this one. Thanks for responding!

Notice that you can define $\omega(x)$ as $$ \frac{x - a}{b - a} = \frac{\omega(x) - 0}{1 - 0} = \omega(x) \Rightarrow x(\omega) = a + (b-a)\omega, $$ and then $$ z(\omega) = \omega - \frac{y(x(\omega)) - \alpha}{\beta - \alpha}. $$

Finally, perform the respective derivatives using this new variables. The differentiation chain rule will be your best friend in that deal!