How do I go about finding an eigenfunction expansion for the following equation:
$$ u'' = f(x)$$ where: $$ u'(0) = \alpha \quad u'(1) = \beta$$
What about the case when $f(x) = C$ a constant?
Edit:
Okay, so I found the eigenfunction:
$$ \phi - \lambda \phi = 0 $$ $$ \rightarrow r^{2} - \lambda = 0 \Rightarrow r = \{ \lambda^{0.5}, -\lambda^{0.5} \}$$
So our general solution is:
$$ \phi = A e^{\sqrt{ \lambda}x} + B e^{-\sqrt{\lambda}x}$$
But, 1. How do I find A and B? and 2. How do I write the eigenfunction expansion? What does that even mean?
Edit: Related, but unsatisfactory Eigenfunction expansion