Ordinary Differential Equation--System with Constant Coefficients
المؤلف:
المرجع الالكتروني للمعلوماتيه
المصدر:
المرجع الالكتروني للمعلوماتيه
الجزء والصفحة:
...
3-7-2018
1093
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 homogeneous case with 
. The solutions to
 |
(2)
|
are given by
 |
(3)
|
But, by the eigen decomposition theorem, the matrix exponential can be written as
 |
(4)
|
where the eigenvector matrix is
![u=[u_1 ... u_n]](http://mathworld.wolfram.com/images/equations/OrdinaryDifferentialEquationSystemwithConstantCoefficients/NumberedEquation5.gif) |
(5)
|
and the eigenvalue matrix is
![D=[e^(lambda_1t) 0 ... 0; 0 e^(lambda_2t) ... 0; | | ... 0; 0 0 ... e^(lambda_nt)].](http://mathworld.wolfram.com/images/equations/OrdinaryDifferentialEquationSystemwithConstantCoefficients/NumberedEquation6.gif) |
(6)
|
Now consider
The individual solutions are then
 |
(10)
|
so the homogeneous solution is
 |
(11)
|
where the 
s are arbitrary constants.
The general procedure is therefore
1. Find the eigenvalues of the matrix 
(
, ..., 
) by solving the characteristic equation.
2. Determine the corresponding eigenvectors 
, ..., 
.
3. Compute
 |
(12)
|
for 
, ..., 
. Then the vectors 
which are real are solutions to the homogeneous equation. If 
is a 
matrix, the complexvectors 
correspond to real solutions to the homogeneous equation given by ![R[x_j]](http://mathworld.wolfram.com/images/equations/OrdinaryDifferentialEquationSystemwithConstantCoefficients/Inline26.gif)
and ![I[x_j]](http://mathworld.wolfram.com/images/equations/OrdinaryDifferentialEquationSystemwithConstantCoefficients/Inline27.gif)
.
4. If the equation is nonhomogeneous, find the particular solution given by
 |
(13)
|
where the matrix 
is defined by
![X(t)=[x_1 ... x_n].](http://mathworld.wolfram.com/images/equations/OrdinaryDifferentialEquationSystemwithConstantCoefficients/NumberedEquation11.gif) |
(14)
|
If the equation is homogeneous so that 
, then look for a solution of the form
 |
(15)
|
This leads to an equation
 |
(16)
|
so 
is an eigenvector and 
an eigenvalue.
5. The general solution is
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