We have the second order differential equation
$\epsilon \dfrac{d^{2}y}{dx^{2}} + \dfrac{dy}{dx} +y = 0$
with boundary values $y(0)=0,\, \, \, y(1)=1$.
I would like to get the exact solution in the form $$y(x) = C \exp(\alpha x)\sinh(\beta x)$$ with $\alpha, \beta$ and $C$ as constants.
I'm too sure how to go about this, I have tried to substitute the solution form into the differential equation but I don't think I am going in the right direction.
Suppose $\epsilon \in (0,1/4)$. The general solution to your equation is linear combinations of $e^{r_1 x}$ and $e^{r_2 x}$ where $r_1,r_2$ solve $\epsilon r^2+r+1=0$. So your solution will be
$$c_1 e^{r_1 x} + c_2 e^{r_2 x}$$
where
$$c_1+c_2=0 \\ c_1 e^{r_1} + c_2 e^{r_2} = 1.$$
It is probably best to solve this system directly, rather than in the simplified form that you want. To actually get to your simplified form, first notice that the first equation tells you $c_1=-c_2$. Second, solve $r_1=\alpha-\beta,r_2=\alpha+\beta$. Then
$$c_1 e^{r_1 x} + c_2 e^{r_2 x} = c_1 e^{\alpha x} \left ( e^{-\beta x} - e^{\beta x} \right ) = -2 c_1 e^{\alpha x} \sinh(\beta x).$$