“Ordinary Differential Equation--System with Constant Coefficients”的概念、定义、习题解题方法及翻译-数学辞典

单词

Ordinary Differential Equation--System with Constant Coefficients

释义

Ordinary Differential Equation--System with Constant Coefficients

To solve the system of differential equations

(1)

where

is a Matrix and

and

are Vectors, first consider the homogeneouscase with

. Then the solutions to

(2)

are given by

(3)

But, by the Matrix Decomposition Theorem, the Matrix Exponential can be written as

(4)

where the Eigenvector Matrix is

(5)

and the Eigenvalue Matrix is

(6)

Now consider



			(7)

The individual solutions are then

(8)

so the homogeneous solution is

(9)

where the

s are arbitrary constants.

The general procedure is therefore

1. Find the Eigenvalues of the Matrix

(

, ...,

) bysolving the Characteristic Equation.

2. Determine the corresponding Eigenvectors

, ...,

3. Compute

(10)

for

, ...,

. Then the Vectors

which are Real aresolutions to the homogeneous equation. If

is a

matrix, the Complex vectors

correspond to Real solutions to the homogeneous equation given by

and

4. If the equation is nonhomogeneous, find the particular solution given by

(11)

where the Matrix

is defined by

(12)

If the equation is homogeneous so that

, then look for a solution of the form

(13)

This leads to an equation

(14)

is an Eigenvector and

an Eigenvalue.

5. The general solution is

(15)

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