What is the significance of the constant m in the Hermite differential equation?

In summary, the conversation is discussing the Hermite's differential equation and the value of m in the equation. It is mentioned that m is usually a nonnegative integer and that there exists a polynomial solution when m = 2n, where n is a nonnegative integer. The conversation also references a website for further analysis and information on the topic.
  • #1
Mrinmoy Naskar
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y" - 2xy' + my = 0 this is well known hermite diff eqn. now can anyone tell me what kind of conts is m?? what is the suitable value of m??
 
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  • #2
Mrinmoy Naskar said:
y" - 2xy' + ny = 0 this is well known hermite diff eqn. now can anyone tell me what kind of conts is m?? what is the suitable value of m??

please explain what is 'm'?
 
  • #3
drvrm said:
please explain what is 'm'?
sorry for the confusion.. I made the correction in my qus...
 
  • #4
Mrinmoy Naskar said:
y" - 2xy' + my = 0 this is well known hermite diff eqn. now can anyone tell me what kind of conts is m?? what is the suitable value of m??
Several web pages that I looked at say that m is usually a nonnegative integer. Did you try searching for yourself?
 
  • #5
Mrinmoy Naskar said:
can anyone tell me what kind of conts is m?? what is the suitable value of m??

This is Hermite’s equation, where special choices of m give rise to the Hermite polynomials.
there exists a polynomial solution of the original equation whenever m = 2n , n = 0,1,2 , , , ... . With the choice m = 2n , and the arbitrary multiplicative constant chosen so that the coefficient of the term x^n is 2^(m/2)
 
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  • #6
Mark44 said:
Several web pages that I looked at say that m is usually a nonnegative integer. Did you try searching for yourself?
@mrinmoy Pl. see a detail analysis
in < http://www.ncl.ac.uk/maths/students/teaching/notebooks/SeriesSolnNotebook.pdf>
 
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Related to What is the significance of the constant m in the Hermite differential equation?

What is the Hermite differential equation?

The Hermite differential equation is a second-order linear differential equation that is commonly used in mathematical physics. It is named after the French mathematician Charles Hermite and has the form y'' - 2xy' + 2ny = 0, where n is a constant.

What is the significance of the Hermite differential equation?

The Hermite differential equation has many applications in physics, specifically in quantum mechanics and statistical mechanics. It is used to describe the behavior of quantum harmonic oscillators and the energy levels of certain physical systems.

How is the Hermite differential equation solved?

The Hermite differential equation can be solved using various methods, including power series, Frobenius method, and the Heun method. The solution involves finding the eigenvalues and eigenfunctions of the differential equation, which are used to construct the general solution.

What are some real-world examples of the Hermite differential equation?

The Hermite differential equation can be used to model the vibrations of diatomic molecules, the motion of a charged particle in a magnetic field, and the behavior of electrons in a crystal lattice. It is also used in statistics to describe the probability distribution of a normal random variable.

Are there any variations of the Hermite differential equation?

Yes, there are various forms of the Hermite differential equation, such as the modified Hermite equation and the generalized Hermite equation. These variations have different coefficients and may have additional terms, but they still follow the basic structure of a second-order linear differential equation.

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