# Getting Eigenvalues Into a Differential Operator

1. Oct 6, 2013

### bolbteppa

Following Butkov, a second order ode

$$A(x)y'' + B(x)y' + C(x)y = D(x)$$

can always be brought into Sturm-Liouville form

$$\tfrac{d}{dx}[p(x)y'] - s(x)y = f(x)$$

after multiplying across by

$$H(x) = - \tfrac{1}{A(x)}e^{\int^x \tfrac{B(t)}{A(t)}dt}.$$

He then says the function $s(x)$ can "often" be written as

$$s(x) = s_0(x) - \lambda r_0(x)$$

where $0 \leq s_0(x)$, $0 \leq r_0(x)$ & $\lambda$ is fixed.

I just don't see how once can be comfortable with this or how one can use a statement like this in general. How does one take a general second order ode & concretely turn it into something involving $\lambda$?

For instance, in the case that $A$, $B$ & $C$ are polynomials of degree $2$, $1$ & $0$:

$$(ax^2+bx+c)y'' + (dx + e)y' + fy = F(x)$$

I can see that $f$ will be an eigenvalue after multiplication by $H(x)$, but how would you deal with a case like

$$(ax^2+bx+c)y'' + (dx + e)y' + \sin(x)y = F(x)$$