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

单词

Ordinary Differential Equation--First-Order

释义

Ordinary Differential Equation--First-Order

Given a first-order Ordinary Differential Equation

(1)

can be expressed using Separation of Variables as

(2)

then the equation can be expressed as

(3)

and the equation can be solved by integrating both sides to obtain

(4)

Any first-order ODE of the form

(5)

can be solved by finding an Integrating Factor

such that

(6)

Dividing through by

yields

(7)

However, this condition enables us to explicitly determine the appropriate

for arbitrary

and

. To accomplishthis, take

(8)

in the above equation, from which we recover the original equation (5), as required, in the form

(9)

But we can integrate both sides of (8) to obtain

(10)

(11)

Now integrating both sides of (6) gives

(12)

(with

now a known function), which can be solved for

to obtain

(13)

where

is an arbitrary constant of integration.

Given an th-order linear ODE with constant Coefficients

(14)

first solve the characteristic equation obtained by writing

(15)

and setting

to obtain the

Complex Roots.

(16)

(17)

Factoring gives the Roots

(18)

For a nonrepeated Real Root

, the corresponding solution is

(19)

If a Real Root

is repeated

times, the solutions are degenerate and the linearlyindependent solutions are

(20)

Complex Roots always come in Complex Conjugate pairs,

. For nonrepeatedComplex Roots, the solutions are

(21)

If the Complex Roots are repeated

times, the linearly independent solutionsare

(22)

Linearly combining solutions of the appropriate types with arbitrary multiplicative constants then gives the completesolution. If initial conditions are specified, the constants can be explicitly determined. For example, consider thesixth-order linear ODE