4.1we - 4.1: GENERAL THEORY OF HIGHER ORDER LINEAR...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 4.1: GENERAL THEORY OF HIGHER ORDER LINEAR EQUATIONS KIAM HEONG KWA An n th order linear differential equation is an equation of the form (1) P ( t ) d n y dt n + P 1 ( t ) d n- 1 y dt n- 1 + + P n- 1 ( t ) dy dt + P n ( t ) y = G ( t ) , where P i ( t ), i = 0 , 1 , ,n , and G ( t ) are continuous functions on an open interval I . If P ( t ) 6 = 0 at a point t I , then there is a maximal open subinterval J I that contains t and in which P ( t ) is nowhere zero. Hence with no loss of generality, we may assume that P ( t ) 6 = 0 everywhere on I . In this case, (1) is equivalent to the equation (2) d n y dt n + p 1 ( t ) d n- 1 y dt n- 1 + + p n- 1 ( t ) dy dt + p n ( t ) y = g ( t ) on I , where p i ( t ) = P i ( t ) /P , i = 1 , 2 , ,n , and g ( t ) = G ( t ) P ( t ) . If the function g ( t ) is not the zero function on I , then (2) is called a non- homogeneous equation with its associated homogeneous equation given by (3) d n y dt n + p 1 ( t ) d n- 1 y dt n- 1 + + p n- 1 ( t ) dy dt + p n ( t ) y = 0 . We shall devote most of this section to outline the general theory of (3). As a fundamental result in this respect, we have Theorem 1 (Existence and Uniqueness Theorem) . If the coefficient functions p i ( t ) , i = 1 , 2 ,n , and g ( t ) in (2) are continuous on an open interval I that contains a point t , then there exists a unique solution y ( t ) of (2) , defined on I , that satisfies the initial conditions (4) y ( t ) = y , y ( t ) = y , , y ( n- 2) ( t ) = y ( n- 2) , y ( n- 1) ( t ) = y ( n- 1) for any prescribed constants y , y , , y ( n- 2) , and y ( n- 1) . Date : January 26, 2011. 1 2 KIAM HEONG KWA The Determinant of a Square Matrix. To generalize the idea of the Wronskian, we need to define the determinant of a square matrix. A square matrix A = ( a ij ) of order n consists of an n-by- n array of n 2 scalars, called the entries of A , arranged as follows: A = a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . . . . . a n 1 a n 2 a nn . The determinant of a square matrix is defined inductively as follows....
View Full Document

Page1 / 11

4.1we - 4.1: GENERAL THEORY OF HIGHER ORDER LINEAR...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online