{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# L94 - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

This preview shows pages 1–2. Sign up to view the full content.

Fall 2003 Math 308/501–502 9 Matrix Methods for Linear Systems 9.4 Linear Systems in Normal Form Wed, 12/Nov c ± 2003, Art Belmonte Summary Deﬁnition of a linear system; matrix notation A linear system is one that may be written in the normal form x 0 ( t ) = Ax ( t ) + f ( t ) or more brieﬂy, x 0 = Ax + f .Here x is an n × 1 column vector function, A is an n × n matrix function, and f is an n × 1 column vector function (the forcing term ). If f = 0 ,the n × 1 zero vector function, then the system is homogeneous ; otherwise it is said to be nonhomogeneous . The j -th row in this vector differential equation is x 0 j = f j ( t ) + n ± k = 1 a jk ( t ) x k ( t ). The sum on the right-hand side consists of j -th element of the forcing term and the dot product of the j -th row of A with the column vector x . Note that the x k appear solely to the ﬁrst power (hence the phrase “linear”). Moreover, while the a and f k functions may depend on the independent variable t (or be constants), they do not depend on the dependent variables (the x k ). YALO (“Yet Another Linear Operator”) Rewrite x 0 = Ax + f as x 0 - Ax = f and let L [ x ] = x 0 - Ax . Then the system becomes L [ x ] = f . (NOTE: In this form, it’s actually easier to see why the linear system is called homogeneous if f = 0 and nonhomogeneous if f 6= 0 .) That L is a linear operator follows immediately from the linearity of differentiation and matrix muliplication. Initial value problem for a linear system of ODEs This consists of the differential equation x 0 = Ax + f together with the initial condition x ( t 0 ) = x 0 . Existence and Uniqueness Theorem With A ( t ) and f ( t ) deﬁned as above and continuous on an interval I ,let t 0 I and x 0 R n . Then the initial value problem x 0 ( t ) = A ( t ) x ( t ) + f ( t ), x ( t 0 ) = x 0 has a unique solution deﬁned for all t I . Converting higher-order linear ODEs to systems Given the n th order linear equation in standard form y ( n ) ( t ) + n - 1 ± j = 0 p j ( t ) y ( j ) ( t ) = g ( t ), let x 1 = y , x 2 = y 0 , x 3 = y 00 , ... , x n = y ( n - 1 ) . We thus have x 0 k = y ( k ) = x k + 1 , k = 1 , 2 ,... n - 1. Moreover, x 0 n = y ( n ) = g ( t ) - n - 1 ± j = 0 p j ( t ) y ( j ) ( t ). Therefore, we have the linear system x 0 1 = x 2 x 0 2 = x 3 ··· x 0 n - 1 = x n x 0 n = g ( t ) - n ± j = 1 p j ( t ) y ( j ) ( t ).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

L94 - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online