I want to consider a general BVP equation, and then convert it to a Fredholm integral equation so that any other similar problems cause less chaos.
Choosing the equation:
$y''+p(x)y'+q(x)y=f(x) $
(where $\,a\leq x\leq b\,$)
subject to the following conditions:
$ \,y(a)=\alpha\, $ and $\,y(b)=\beta$.
Can anyone help out, or walk me through it. Thanks. I lost it when i had to integrate the second time. I tried looking around but no examples cover this general case. I am sure after this example i will have less or no trouble with BVP <==> Fredholm. hopefully. If a similar question or somewhat close question was posted before, a link will suffice, but i could not find any. Thanks all.
PS. Why do we make these conversions?
Assuming that $y'' = u(x)$, successively integrating both sides to obtain the values of $y'(x)$ and $y(x)$ over the given interval did the trick. That's the algorithm i used, and following that in an organised manner simplified everything beyond belief :) !