Using the solution method used in my answer to this question. Deal with the case where $\alpha = \beta$. So we start with the equation $$v(a + 1) = v(a) + v(1)$$
I'm not sure what I need to prove next. Am I suppose to prove by induction that $$v(a) = c + da,$$ and then use the boundary conditions $v(0) = 0, v(m+n) = 1$ to show that $$v(a) = \frac{a}{m+n}$$?
