My course's notes say that for a function $u=u(x,y)$, if we define $\alpha = x+ay$, and $\beta = x+by$, then:
$u_x = u_\alpha + u_\beta$, and $u_y = au_\alpha + bu_\beta$.
And this can help solve the PDE.
However I fail to see the reasoning for why the above holds.
I know of the chain rule's multivariable case, e.g. for $f(x(t), y(t))$, then $f_t = f_x x_t + f_y y_t$. But I can't seem to apply this to the case above?
You have the chain rule formula right there and you can't apply it?
\begin{align} u_x &= u_\alpha \alpha_x + u_\beta \beta_x = u_\alpha + u_\beta \\ u_y &= u_\alpha \alpha_y + u_\beta \beta_y = a u_\alpha + bu_\beta \end{align}
Since $\alpha_x = \beta_x = 1$, $\alpha_y = a$, $\beta_y = b$