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M334Lec33

M334Lec33 - Math 334 Lecture#33 7.7 Fundamental Matrices...

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Math 334 Lecture #33 § 7.7: Fundamental Matrices Example. Recall that a general solution of x = Ax where A = 1 1 4 1 is x ( t ) = c 1 x (1) + c 2 x (2) = c 1 e 3 t 2 e 3 t + c 2 e - t - 2 e - t = e 3 t e - t 2 e 3 t - 2 e - t c 1 c 2 . The form of this general solution is that of an invertible (why?) 2 × 2 matrix function times a vector of arbitrary constants, x ( t ) = Ψ( t ) c, where the columns of the 2 × 2 the matrix function, Ψ( t ) = e 3 t e - t 2 e 3 t - 2 e - t , are the linearly independent solutions x (1) and x (2) of the system. The function Ψ( t ) is called a fundamental matrix for the system; it satisfies the first order differential matrix equation Ψ ( t ) = A Ψ( t ) where A = 1 1 4 1 . This is true because each column of Ψ( t ) is a solution of the system. This can be verified: Ψ ( t ) = 3 e 3 t - e - t 6 e 3 t 2 e - t and 1 1 4 1 Ψ( t ) = 1 1 4 1 e 3 t e - t 2 e 3 t - 2 e - t = 3 e 3 t - e - t 6 e 3 t 2 e - t . The Form of General Solutions. If x (1) ( t ) , . . . , x ( n ) ( t ) are n linearly independent solutions of x = P ( t ) x where P ( t ) is an n × n matrix function, then a general solution of the system is x ( t

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M334Lec33 - Math 334 Lecture#33 7.7 Fundamental Matrices...

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