Is This a Generalized Form of Legendre's Differential Equation?

[ognik]

The ODE is $ \d{}{x}[(1-x^2)\d{u}{x}]+\alpha u + \beta x^2u = 0 $. I know of Legendre's ODE, and Bessles and a few others - does this one also have a name?

 

[jvicens]

Hello,

Thank you for your question. The ODE in question is known as the Legendre's Differential Equation with a generalization term. It is a second-order linear differential equation with variable coefficients. The general form of this equation is

$$
\d{}{x}[(1-x^2)\d{u}{x}]+\alpha u + \beta x^2u = 0
$$

This equation is named after the French mathematician Adrien-Marie Legendre, who studied it extensively in the late 18th and early 19th century. It has a wide range of applications in physics, engineering, and mathematics, especially in the study of orthogonal polynomials and spherical harmonics.

In addition to the Legendre's ODE, there are other related equations, such as the Bessel's differential equation, which also have a generalization term. These equations are all part of the family of special functions known as the "classical orthogonal polynomials."

I hope this answer helps clarify the name and significance of the ODE in question. Please let me know if you have any further questions.