Getting Eigenvalues Into a Differential Operator

[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


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

 

[jvicens]

Thank you for your post and for bringing up an interesting question. I understand your confusion about the statement made by Butkov regarding the Sturm-Liouville form and the function s(x). Let me try to explain it in a more concrete way.

First, it is important to understand that the Sturm-Liouville form is a specific form of a second-order differential equation. It is often used in mathematical physics and describes a wide range of phenomena, from vibrations of a string to heat diffusion. This form is particularly useful because it can be solved analytically in many cases.

To bring a second-order differential equation into Sturm-Liouville form, we follow the steps outlined by Butkov. We multiply the equation by H(x) and then rewrite it in terms of the Sturm-Liouville operator, which is given by:

L = \tfrac{d}{dx}[p(x)\tfrac{d}{dx}] - s(x)

where p(x) and s(x) are functions defined by:

p(x) = \tfrac{1}{A(x)}e^{\int^x \tfrac{B(t)}{A(t)}dt}

s(x) = \tfrac{C(x)}{A(x)}e^{\int^x \tfrac{B(t)}{A(t)}dt}

Now, let's focus on the function s(x). As Butkov mentioned, it can often be written as s(x) = s_0(x) - \lambda r_0(x), where s_0(x) and r_0(x) are non-negative functions and \lambda is a fixed parameter. This form is useful because it allows us to solve the Sturm-Liouville equation by finding the eigenvalues and eigenfunctions of the operator L.

To see how this works, let's take a concrete example. Consider the following differential equation:

y'' + x^2y = 0

We can bring this into Sturm-Liouville form by multiplying it by H(x) = -\tfrac{1}{1}e^{\int^x x^2dx} = -e^{\tfrac{x^3}{3}}. This gives us:

\tfrac{d}{dx}[e^{\tfrac{x^3}{3}}y'] - x^2y = 0

Now, we can rewrite this in terms of the Sturm-Liou