Read More
Date: 21-5-2018
![]()
Date: 5-7-2018
![]()
Date: 22-6-2018
![]() |
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
![]() |
(5) |
and the eigenvalue matrix is
![]() |
(6) |
Now consider
![]() |
![]() |
![]() |
(7) |
![]() |
![]() |
![]() |
(8) |
![]() |
![]() |
![]() |
(9) |
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
and
.
4. If the equation is nonhomogeneous, find the particular solution given by
![]() |
(13) |
where the matrix is defined by
![]() |
(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
![]() |
|
|
منها نحت القوام.. ازدياد إقبال الرجال على عمليات التجميل
|
|
|
|
|
دراسة: الذكاء الاصطناعي يتفوق على البشر في مراقبة القلب
|
|
|
|
|
هيئة الصحة والتعليم الطبي في العتبة الحسينية تحقق تقدما بارزا في تدريب الكوادر الطبية في العراق
|
|
|