I want to study the solutions of the following boundary value problem:
$$\begin{cases} -x''(t) +p(t)x'(t)+q(t)x(t)=r(t),\, t \in [t_0, T]\\ x(t_0) = \alpha\\ \mu x(T) + x'(T) = \beta \end{cases}$$
What I need is an auxiliary problem $-y''(t)+p(t)y'(t)+q(t)y(t)=r(t),\, t\in [t_0, T]$ such that its solution $y(t)$ is expressible as a combination $$y(t) = y_1(t) + sy_2(t)$$ but I don't know what the initial conditions should be. Any help with that would be highly appreciated. Thank you.
The differential equation being linear and inhomogeneous, its general solution is $x(t) = x_0(t) + a x_1(t) + b x_2(t)$, where $x_0$ is one particular solution, $x_1$ and $x_2$ are a fundamental set of solutions of the homogeneous equation $-x''(t) + p(t) x'(t) + q(t) x(t) = 0$, and $a$ and $b$ are arbitrary constants. The boundary conditions give you a system of two equations in $a$ and $b$:
$$ \eqalign{ a x_1(t_0) + b x_2(t_0) &= \alpha - x_0(t_0)\cr a (\mu x_1(T) + x_1'(T)) + b (\mu x_2(T) + x_2'(T)) &= \beta - (x_1(T) + x_2'(T))\cr} $$
The system may or may not be nonsingular. If it is, there is a unique solution to the boundary value problem. If not, either there is no solution or there are infinitely many.
EDIT: In the "shooting" method, you might take $y_1(t)$ to be a particular solution of the non-homogeneous equation with $y_1(t_0) = \alpha$, and $y_2(t)$ a solution of the homogeneous equation with $y_2(t_0) = 0$ but $y_2'(t_0) \ne 0$.