7.4 - 7.4: BASIC THEORY OF SYSTEMS OF FIRST ORDER LINEAR...

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7.4: BASIC THEORY OF SYSTEMS OF FIRST ORDER LINEAR EQUATIONS KIAM HEONG KWA 1. The Wronskian of Solutions and The Liouville’s Formula Consider the homogeneous system of n first order linear equations (1.1) x 0 = P ( t ) x , where (1.2) x = x 1 x 2 . . . x n and P ( t ) = p 11 ( t ) p 12 ( t ) ··· p 1 n ( t ) p 21 ( t ) p 22 ( t ) ··· p 2 n ( t ) . . . . . . . . . . . . p n 1 ( t ) p n 2 ( t ) ··· p nn ( t ) , the latter of which is assumed to be continuous in an open interval I . More explicitly, one has x 0 1 = p 11 x 1 + p 12 ( t ) x 2 + ··· + p 1 n ( t ) x n , x 0 2 = p 21 x 1 + p 22 ( t ) x 2 + ··· + p 2 n ( t ) x n , . . . x 0 n = p n 1 x 1 + p n 2 ( t ) x 2 + ··· + p nn ( t ) x n . Let (1.3) x (1) ( t ) = x 11 ( t ) x 21 ( t ) . . . x n 1 ( t ) , x (2) ( t ) = x 12 ( t ) x 22 ( t ) . . . x n 2 ( t ) , ··· , x ( j ) ( t ) = x 1 j ( t ) x 2 j ( t ) . . . x nj ( t ) , ··· , x ( n ) ( t ) = x 1 n ( t ) x 2 n ( t ) . . . x nn ( t ) Date : March 1, 2011. 1
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KIAM HEONG KWA be n solutions of (1.1). Then (1.4) Ψ ( t ) = ( x (1) ( t ) x (2) ( t ) ··· x ( n ) ( t ) ) = x 11 ( t ) x 12 ( t ) ··· x 1 n ( t ) x 21 ( t ) x 22 ( t ) ··· x 2 n ( t ) . . . . . . . . . . . . x n 1 ( t ) x n 2 ( t ) ··· x nn ( t ) is a matrix solution of (1.1) in the sense that (1.5) Ψ 0 ( t ) = P ( t ) Ψ ( t ) . The Wronskian of x (1) ( t ) , x (2) ( t ) , ··· , x ( n ) ( t ) is the function (1.6) W ( t ) = W [ x (1) , x (2) , ··· , x ( n ) ]( t ) := det Ψ ( t ) . The goal of this section is to show that the Wronskian W ( t ) satisfies the Liouville’s formula (1.7) dW dt = [tr P ( t )] W. The proof is straightforward in the case where n = 2. In this case, since W ( t ) = x 11 ( t ) x 22 ( t ) - x 12 ( t ) x 21 ( t ), W 0 ( t ) = x 0 11 ( t ) x 22 ( t ) - x 0 12 ( t ) x 21 ( t ) + x 11 ( t ) x 0 22 ( t ) - x 12 ( t ) x 0 21 ( t ) . On the other hand, since d x ( j ) dt = P ( t ) x ( j ) , it follows that x 0 ij ( t ) = 2 X k =1 p ik ( t ) x kj ( t ) for i,j = 1 , 2. Substituting these equations into the one for W 0 ( t ) yields W 0 ( t ) = 2 X k =1 [ p 1 k ( t ) x k 1 ( t ) x 22 ( t ) - p 1 k ( t ) x k 2 ( t ) x 21 ( t ) + x 11 ( t ) p 2 k ( t ) x k 2 ( t ) - x 12 ( t ) p 2 k ( t ) x k 1 ( t )] = 2 X k =1 { p 1 k ( t ) [ x k 1 ( t ) x 22 ( t ) - x k 2 ( t ) x 21 ( t )] + p 2 k ( t ) [ x 11 ( t ) x k 2 ( t ) - x 12 ( t ) x k 1 ( t )] = 2 X k =1 p kk ( t ) [ x 11 ( t ) x 22 ( t ) -
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7.4 - 7.4: BASIC THEORY OF SYSTEMS OF FIRST ORDER LINEAR...

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