Sturm-Liouville Theory Question

[N00813]

Homework Statement

Given the ODE:
$$(1-x^2) \frac{d^2y}{d^2x} - x \frac{dy}{dx} + n^2 y = 0$$

and that the weight function of the differential operator $$-(1-x^2) \frac{d^2y}{d^2x} + x \frac{dy}{dx}$$ is $$w = \frac{1}{\sqrt{1-x^2}}$$ to turn it into an SL operator, prove that:

$$\int_{-1}^{1} \frac{dy_m}{dx} \frac{dy_n}{dx} \sqrt{1-x^2} dx = 0$$

where $$y_m$$ and $$y_n$$ are orthogonal eigenfunctions of the ODE.

Homework Equations

All above.

The Attempt at a Solution

I attempted integration by parts of the LHS of the proposition, but that didn't seem to go anywhere.

 

[vela]

Integration by parts worked for me.

 

[N00813]

Integration by parts worked for me.

I tried integrating one of the derivatives, and differentiating everything else, but this ended with a 2nd derivative multiplied with a quotient.

I also tried integrating 1 and differentiating the integrand. To get rid of the second derivatives, I use the ODE to substitute, but that still gave me a quotient term I couldn't get rid of.

Can you point me in a specific direction?

 

[vela]

Not really other than to say do the integration by parts correctly. Apparently you're doing something wrong or not seeing something, but without seeing your actual work, no one can say what it is. Vague descriptions aren't very helpful to us.

 

[N00813]

Not really other than to say do the integration by parts correctly. Apparently you're doing something wrong or not seeing something, but without seeing your actual work, no one can say what it is. Vague descriptions aren't very helpful to us.

How did you start off, then? Which part of the integrand did you choose to integrate and differentiate?

 

[pasmith]

How did you start off, then? Which part of the integrand did you choose to integrate and differentiate?

Clearly one doesn't want to integrate $y_n'\sqrt{1 - x^2}$, so one must differentiate $y_n'\sqrt{1 - x^2}$ and integrate $y_m'$. You want to end up with a multiple of
$$\int_{-1}^1 y_n(x) y_m(x) w(x)\,dx = \int_{-1}^1 \frac{y_n(x) y_m(x)}{\sqrt{1 - x^2}}\,dx$$
which is zero if $y_n$ and $y_m$ are orthogonal.

 

[N00813]

Ah, thanks.

Finally figured it out!