“Hermite Differential Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Hermite Differential Equation

释义

Hermite Differential Equation

(1)

This differential equation has an irregular singularity at

. It can be solved using the series method

(2)

(3)

Therefore,

(4)

and

(5)

for

, 2, .... Since (4) is just a special case of (5),

(6)

for

, 1, .... The linearly independent solutions are then

, 4, 8, ..., then

terminates with the Power

, and

(normalized so thatthe Coefficient of

) is the regular solution to the equation, known as the Hermite Polynomial. If

, 6, 10, ..., then

terminates with the Power

, and

(normalized sothat the Coefficient of

) is the regular solution to the equation, known as the Hermite Polynomial.

If , then Hermite's differential equation becomes

(9)

which is of the form

and so has solution

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