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Unformatted text preview: CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 10 Posted Wednesday 19 March 2008. Due Wednesday 26 March 2008 in class. 1. [40 points] For each of the following examples, (1) compute the matrix exponential e t A by hand, and (2) explain the behavior of d x /dt = Ax as t given that x (0) = (2 , 0) T . ( a ) A = parenleftbigg 1 1 parenrightbigg ( b ) A = parenleftbigg 50 49 49 50 parenrightbigg ( c ) A = parenleftbigg 1 1 parenrightbigg . Hint for part (c): This matrix is not symmetric, but it does have a decomposition of of the form A = UDU * , where U is a matrix of (possibly complex) eigenvectors, D is a diagonal matrix of eigenvalues, and U * is the transpose of the complex conjugate of U . 2. [30 points] For the matrix A from problem 1(b), describe how large one can choose the time step t so that the forward Euler method applied to d x /dt = Ax , x k +1 = x k + t Ax k , will produce a solution that qualitatively matches the behavior of the true solution (i.e., the approxi-will produce a solution that qualitatively matches the behavior of the true solution (i....
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