“Abel's Identity”的概念、定义、习题解题方法及翻译-数学辞典

单词

Abel's Identity

释义

Abel's Identity

Given a homogeneous linear Second-Order Ordinary Differential Equation,

(1)

call the two linearly independent solutions

and

. Then

(2)

(3)

Now, take

(3) minus

(2),

(4)

(5)

(6)

Now, use the definition of the Wronskian and take its Derivative,

			(7)

			(8)

Plugging

and

into (6) gives

(9)

This can be rearranged to yield

(10)

which can then be directly integrated to

(11)

where

is the Natural Logarithm. Exponentiation then yields Abel's identity

(12)

where

is a constant of integration.

See also Ordinary Differential Equation--Second-Order

References

Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, pp. 118, 262, 277, and 355, 1986.

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