I've been trying to solve this equation $$-\mu u'' + \beta u' = 1$$ where $u(0) = 0$, $u'(1) = 1$. So far the result I have is $$u = \frac{(exp(\frac{\beta x}{\mu})-1)\mu}{\beta exp(\frac{\beta}{\mu})}$$ but when I tried to substitute it on the equation the result is not the same.
What am I missing? I know that to get the solution we have to find two "solutions" the general and the specific one, can anyone explain how to solve this?