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

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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 0 ( 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 0 ( t 0 ) 6 = 0 at a point t 0 I , then there is a maximal open subinterval J I that contains t 0 and in which P 0 ( 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 0 , i = 1 , 2 , · · · , n , and g ( t ) = G ( t ) P 0 ( 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 0 , then there exists a unique solution y ( t ) of (2) , defined on I , that satisfies the initial conditions (4) y ( t 0 ) = y 0 , y 0 ( t 0 ) = y 0 0 , · · · , y ( n - 2) ( t 0 ) = y ( n - 2) 0 , y ( n - 1) ( t 0 ) = y ( n - 1) 0 for any prescribed constants y 0 , y 0 0 , · · · , y ( n - 2) 0 , and y ( n - 1) 0 . Date : January 26, 2011. 1

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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. The determinant of a square matrix that consists of a single entry is defined to be the value of the single entry. For a square matrix A = ( a ij ) of order n , where n 2, its determinant is defined by (5) det( A ) = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n · · · · · · · · · · · · a n 1 a n 2 · · · a nn = n X j =1 ( - 1) i + j a ij det( A ij ) , where A ij is the square matrix of order n -
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4.1we - 4.1 GENERAL THEORY OF HIGHER ORDER LINEAR EQUATIONS...

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