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Unformatted text preview: MA366 Sathaye Fast methods for some chapter 7 problems. We show some faster methods to quickly get a fundamental matrix for linear systems of differential equations. Two variable systems. 1. Basic formulas. We consider a system X = AX where A is a 2 2 matrix with constant coefficients. Let A = a b c d . We note that its characteristic polynomial has the formula P A ( ) = 2- ( a + d ) + ad- bc . We typically solve for eigenvalues by finding the roots of P A ( ) and depending on the nature of roots, we have different techniques for solving the system. 2. Fundamental Matrix For this discussion, we may even consider an n n matrix A giving a system of n equations in n variables. A Fundamental matrix for the system X = AX is defined to be an n n matrix ( t ) whose columns are n independent solutions of the system. This can be shown equivalent to the condition ( t ) = A ( t ) together with the condition that ( t ) is invertible on an interval where the solutions are discussed. The Special Fundamental Matrix is given by ( t ) = exp ( At ) = I + At + A 2 t 2 / 2! + Its columns are independent solutions of the system and have the added property that (0) = I . Indeed, this extra condition identifies a Fundamental Matrix as a Special Fundamental Matrix. A general solution to the system X = AX is then given by X ( t ) = ( t ) C where C is a column of arbitrary constants. In particular, X = ( t ) X (0) where X (0) = X (0) is the given set of initial conditions. We give direct methods to find this matrix ( t ). 3. Exponential shift. Consider a substitution X = exp( rt ) Y . Substitution in the equation X = AX gives X = exp( rt ) rY + exp( rt ) Y = A exp( rt ) Y. By canceling exp( rt ) and rearranging, we see that Y = ( A- rI ) Y. It is easy to see that the characteristic polynomial of A- rI is simply P A ( + r ). We shall use this substitution to simplify our characteristic polynomial. Finally, whatever special fundamental matrix is found for the system Y = ( a- rI ) Y , we may multiply it by exp( rt ) to get the matrix for the system X = AX ....
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