Stochastic process: boundary condition

Question

Can someone help me with problem 8 in these problem:

I have found the solution in the hint, but how can I use boundary condition in (3) to find a constraint between $C_1$ and $C_2$?

Answer 1

Just substitute the last equation of the image into (3):

$$C_1+C_2\left(\frac{\delta}{\beta}\right)^1=\frac{\beta}{\beta+\delta}\left[C_1+C_2\left(\frac{\delta}{\beta}\right)^2\right]$$

which, if you move all $C_1$ terms to the left and all $C_2$ terms to the right and simplify, leads to

$$C_1=-C_2.$$

This is the constraint you are looking for.