Hi i'm having a bit of trouble working out how to do this question. As i've not done Sturm-Liouville theory before and finding it hard to work it out.
The question is
y''-2y'+(1+lamda)y=0 with homogeneous bc on x=0,1
Write in S-L form
Write the orthagonality condition.
Now i'm having trouble working out how this is done could anybody be kind enough to go through the steps needed to work this out?
Thanks very much
So we solve the associated equation
Now if we complete the square we get
Now we have 3 cases:
If
we get Now if you impose the boundry conditions you get the trivial solution
If you have we get
now if x=0 we get
Now if x=1 we get
Since the exponential is a 1-1 they are not equal so
and we only get the trivial solution
So finally we get to the last case
we get
So when x=0 we get
so the equation reduces to
When x=1 we get
Now we take the inverse sine of both sides to get
This gives
So your orthogonal basis is