Sturm-Liouville Theory Question

• N00813
In summary, the conversation is about proving a proposition involving an ODE and its weight function. The participants discuss using integration by parts to solve the problem, but encounter difficulties with a quotient term. Eventually, they determine that the key is to differentiate and integrate specific parts of the integrand to arrive at the desired result.
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.

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.

Last edited:
Integration by parts worked for me.

vela said:
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?

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.

vela said:
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?

N00813 said:
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.

1 person
Ah, thanks.

Finally figured it out!

What is Sturm-Liouville Theory?

Sturm-Liouville Theory is a mathematical theory used to solve differential equations with boundary conditions. It is named after mathematicians Jacques Charles François Sturm and Joseph Liouville.

What are the applications of Sturm-Liouville Theory?

Sturm-Liouville Theory has applications in various fields including physics, engineering, and economics. It is used to solve differential equations that arise in these fields and provide solutions for physical phenomena such as heat transfer, wave motion, and quantum mechanics.

What are the key concepts in Sturm-Liouville Theory?

The key concepts in Sturm-Liouville Theory include eigenvalues, eigenfunctions, and orthogonality. These concepts are used to find the solutions to differential equations with boundary conditions.

How is Sturm-Liouville Theory related to the Fourier series?

Sturm-Liouville Theory is closely related to the Fourier series as it provides a way to find the coefficients of the series. The eigenfunctions in Sturm-Liouville Theory are the same as the trigonometric functions used in the Fourier series.

What are the limitations of Sturm-Liouville Theory?

Sturm-Liouville Theory has limitations in its applicability to certain types of differential equations. It is most effective for linear second-order differential equations with homogeneous boundary conditions. Nonlinear and higher-order equations may not have solutions that can be expressed using Sturm-Liouville Theory.

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