Properties of a Sturm-Liouville problem I want to show the following problem is regular.
To show a Sturm-Liovulle problem is regular we need to demonstrate that $y''+\frac{b}{a}y'+\frac{1}{a}(c+\lambda)=0$ where $p(x)=e^{\int \frac{b}{a}\,dx}.$  We then have $[p(x)y']'+[q(x)+r(x)\lambda]y=0.$  Now this is regular if (i) $p,p',r,q$ are all real valued continuous on a finite interval $[a,b]$; (ii) $p(x)>0$ and $r(x)>0$ on $[a,b].$
Consider $x^2y''+xy'+\lambda y=0$ where $x>0$, $y(0)=y(e)=1$, and $y'(0)-2y'(e)=2.$
Dividing through by $x^2$ we get $y''+\frac{1}{x}y'+\frac{\lambda}{x^2}y=0.$  If we let $p(x)=x$  we then have $xy''+y'+\frac{\lambda}{x}y=0.$  This yields us $[xy']'+\frac{\lambda}{x}=0.$  If we let $p(x)=x$, $q(x)=0$. and $r(x)=1/x$, does this show that this problem is regular? Am I missing a step?
 A: 
I want to show the following problem is regular.

Well, the difficulty is that this is not a regular Sturm-Liouville problem, at least not as stated. There is also an issue with the boundary conditions, but we will discuss that in a bit.
Writing the original problem in its Sturm-Liouville form,
$$
{1\over w(x)}\left[p(x){dy\over dx}\right]'+q(x)y+\lambda y(x)=0,\quad a<x<b,\tag{1}
$$
we see $p(x)=x$, $q(x)=0$, $w(x)={1\over x}$. Since the boundary conditions are specified at $x=0$ and $x=e$, the interval here is $0<x<e$. For this problem to be regular, we need $p,q,w,p'$ continuous on $[0,e]$, and $w(x)={1\over x}$ fails this condition. So the problem is not regular.
However, the spirit of the problem can still be salvaged: singular Sturm-Liouville problems aim to relax the hypotheses on the functions $p,q,w$ and/or the interval $[a,b]$ while preserving the important qualitative properties of the solution to the problem. Here's a relevant formulation of this notion applicable to your problem:
(1) is singular if $p,q,w,p'$ are continuous on $a<x<b$, but $p$, $q$, or $w$ becomes infinite at an endpoint.
Since the interval is $0<x<e$, and $w(x)={1\over x}\to\infty$ as $x\to 0^+$, this is a singular S-L problem.
So either you need to appeal to this notion of singular S-L problems or you need to study the problem on an interval away from zero.
Finally, the boundary conditions stated don't make sense since they are specifying three conditions for a second order problem. Maybe you want just $y(0)=y(e)=1$ without the other condition? In such a case, the boundary condition at the singular endpoint is usually replaced by a boundedness condition there, such as 
$$
y(x),y'(x)\text{ bounded as }x\to 0^+.
$$
For the full details of this discussion, a book like Zettl's Sturm-Liouville Theory is appropriate.
