Okay I am having trouble with the question, and before I start, I have not done any working as it how the questions worded and asking for is confusing me, so I will try and convey what I think it telling me, but I just cannot see how the solution given is being formed.
It is part b) I am stuf with as I dont quite understand the statement
"By assuming $B(x)$ gives the behaviour of $\psi(x)$ for $0\leq x <<1$"
at first I was think maybe a binomial expansion but could not see a way of applying that to either the differential or the wave function but I only have one solution for that, that I know of.
I then thought maybe that as x<<1 then as there is a $x^2$ in the denominator then all the other would be invalid this give an second order of
$$y''-\frac{\left(k^2-1\right)}{4x^2}y=0$$ and I am not quite sure how to solve that.
And that all I have come up with so far, I really am not sure how this condition is being applied, to be honest it my first time tackling this kinda of problem, so a nudge in the right direction would be appreciated.
Sorry I should point out it part b) of this problem that is the issue.
also I have just notice how can x=0 and also be x<<1