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Unformatted text preview: Fundamental Matrix Solutions Now that we can solve the homogeneous equation y = A y we will develop some machinery that will become useful as we move toward techniques for solving the nonhomogeneous equation. Recall that the general solution to the homogeneous equation above has the form y ( x ) = c 1 y 1 + c 2 y 2 + + c n y n where y 1 , y 2 ,..., y n are linearly independent solutions. The following definition gives a more concise way of recording this data: Definition Given the homogeneous equation y = A y , a matrix X ( x ) is a fundamental matrix solution to this equation if the columns of X form a set of n linearly-independent solutions. I Example The homogeneous equation y = 1 1- 1 y has eigenvalues 1 = 1 and 2 =- 1 , with corresponding eigenvectors v 1 = 1 and v 2 =- 1 2 . Then we have two linearly-independent solutions y 1 = e x 1 = e x and y 2 = e- x- 1 2 =- e- x 2 e- x so a fundamental matrix solution is given by X = e x e- x 2 e- x ....
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This note was uploaded on 03/03/2012 for the course MATH 4430 at Colorado.