Differential Equations - Ordinary Differential Equations - Homogeneous Linear Systems With Constant Coefficients
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x(1) (t) =
"
e2t e2t e2t
#
,
x(2) (t) =
and x(3) (t) =
"
0 e−t −e−t
#
"
−e−t 0 e−t
#
,
.
It may also be verified that W [x(1) , x(2) , x(3) ] = −3. Hence any solution of the system is given by x = c1 x(1) + c2 x(2) + c3 x(3) and a fundamental matrix is Ψ(t) =
"
e2t e2t e2t
−e−t 0 e−t
0 e−t −e−t
#
.
⊠
§8 Homogeneous linear systems with constant coefficients Here we consider the situation when P(t) is the constant real-valued matrix A. In this section we shall consider only the homogeneous system, that is, x˙ = Ax.
(I.10) OHP 35
To proceed further, we need some linear algebra. Definition. Let A be a square matrix. The scalar r is an eigenvalue of A if there exists a vector w 6= 0 such that Aw = r w. Any such w satisfying this equation is called an eigenvector of A corresponding to r . Theorem I.4. A scalar r is an eigenvalue of an n × n matrix A if and only if det(A − r In ) = 0,
where In is the n × n identity matrix. This theorem shows we can find the eigenvalues by setting up the matrix A − r In , working out its determinant, and then finding the values of r which makes the determinant zero. For each eigenvalue r , we then find the corresponding eigenvector w by solving (A − r In )w = 0. Note that if an eigenvalue occurs m times, the eigenvector w that is found will contain k arbitrary constants, where 1 ≤ k ≤ m. One can then obtain k eigenvectors by making choices of the k parameters. OHP 36
"
e2t e2t e2t
#
,
x(2) (t) =
and x(3) (t) =
"
0 e−t −e−t
#
"
−e−t 0 e−t
#
,
.
It may also be verified that W [x(1) , x(2) , x(3) ] = −3. Hence any solution of the system is given by x = c1 x(1) + c2 x(2) + c3 x(3) and a fundamental matrix is Ψ(t) =
"
e2t e2t e2t
−e−t 0 e−t
0 e−t −e−t
#
.
⊠
§8 Homogeneous linear systems with constant coefficients Here we consider the situation when P(t) is the constant real-valued matrix A. In this section we shall consider only the homogeneous system, that is, x˙ = Ax.
(I.10) OHP 35
To proceed further, we need some linear algebra. Definition. Let A be a square matrix. The scalar r is an eigenvalue of A if there exists a vector w 6= 0 such that Aw = r w. Any such w satisfying this equation is called an eigenvector of A corresponding to r . Theorem I.4. A scalar r is an eigenvalue of an n × n matrix A if and only if det(A − r In ) = 0,
where In is the n × n identity matrix. This theorem shows we can find the eigenvalues by setting up the matrix A − r In , working out its determinant, and then finding the values of r which makes the determinant zero. For each eigenvalue r , we then find the corresponding eigenvector w by solving (A − r In )w = 0. Note that if an eigenvalue occurs m times, the eigenvector w that is found will contain k arbitrary constants, where 1 ≤ k ≤ m. One can then obtain k eigenvectors by making choices of the k parameters. OHP 36
