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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....
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