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

单词

Ordinary Differential Equation Second-Order

释义

Ordinary Differential Equation Second-Order

An ODE

(1)

has singularities for finite

under the following conditions: (a) If either

diverges as

, but

and

remain finite as

, then

is called a regular or nonessentialsingular point. (b) If

diverges faster than

so that

, or

diverges faster than

so that

, then

is called anirregular or essential singularity.

Singularities of equation (1) at infinity are investigated by making the substitution , so , giving

(2)


			(3)

Then (1) becomes

(4)

Case (a): If

			(5)
			(6)

remain finite at

(

), then the point is ordinary. Case (b): If either

diverges no morerapidly than

diverges no more rapidly than

, then the point is a regular singular point. Case (c): Otherwise, the point is an irregular singular point.

Morse and Feshbach (1953, pp. 667-674) give the canonical forms and solutions for second-order ODEs classified bytypes of singular points.

For special classes of second-order linear ordinary differential equations, variable Coefficients canbe transformed into constant Coefficients. Given a second-order linear ODE with variableCoefficients

(7)

Define a function

(8)

(9)

(10)

(11)

This will have constant Coefficients if

and

are not functions of

. But we are free to set

to an arbitrary Positive constant for

by defining

(12)

Then

(13)

(14)

and


			(15)

Equation (11) therefore becomes

(16)

which has constant Coefficients provided that

(17)

Eliminating constants, this gives

(18)

So for an ordinary differential equation in which

is a constant, the solution is given by solving the second-orderlinear ODE with constant Coefficients

(19)

for

, where

is defined as above.

A linear second-order homogeneous differential equation of the general form

(20)

can be transformed into standard form

(21)

with the first-order term eliminated using the substitution

(22)

Then

(23)

(24)

(25)


			(26)

(27)

Therefore,

(28)

where

(29)

If , then the differential equation becomes

(30)

which can be solved by multiplying by

(31)

to obtain

(32)

(33)

(34)

If one solution () to a second-order ODE is known, the other () may be found using the Reduction ofOrder method. From the Abel's Identity

(35)

where

(36)

(37)

(38)

(39)

But

(40)

Combining (39) and (40) yields

(41)

(42)

Disregarding

, since it is simply a multiplicative constant, and the constants

and

, which will contribute asolution which is not linearly independent of

(43)

, this simplifies to

(44)

For a nonhomogeneous second-order ODE in which the term does not appear in the function ,

(45)

let

, then

(46)

So the first-order ODE

(47)

if linear, can be solved for

as a linear first-order ODE. Once the solution is known,

(48)

(49)

On the other hand, if is missing from ,

(50)

let

, then

, and the equation reduces to

(51)

which, if linear, can be solved for

as a linear first-order ODE. Once the solution is known,

(52)

See also Abel's Identity, Adjoint Operator

References

Arfken, G. ``A Second Solution.'' §8.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 467-480, 1985.

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

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 667-674, 1953.

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