L97M - Fall 2003 Math 308/501502 9 Matrix Methods for...

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Fall 2003 Math 308/501–502 9 Matrix Methods for Linear Systems 9.7M Nonhomogeneous Lin Sys: Method of Undetermined Coefficients Mon, 24/Nov c ± 2003, Art Belmonte Summary Method of Undetermined Coefficients for Systems Let A be an n × n real constant matrix and f an n × 1 column vector whose elements are sums of products of real polynomials, sines, cosines, and/or exponentials involving the independent variable t . We may use the method of undetermined coefficients procedure from Section 6.3 as a guide to finding a particular solution x p to the nonhomogeneous linear system x 0 = Ax + f or L [ x ] = x 0 - Ax = f . The undetermined coefficients involved are now symbolic vector constants. Moreover, in case an element of f is replicated in a general solution of the associated homogeneous linear system L [ x ] = 0 , the original choice for a particular solution must not only be multiplied by the smallest positive integer power of t so that no term of the particular solution x p is a solution of the homogeneous equation L [ x ] = 0 , but also by all lower nonnegative integer powers of t as well. This is easier said than done. Indeed, when this level of complexity is reached, it is simpler to resort to variation of parameters , the other technique for finding particular solutions that we encountered. This will be discussed later in lecture handout 9.7V . Superposition Principle For k = 1 , 2 ,..., M ,let x p k be a solution of L [ x ] = f k . Then for any constants c 1 c M , the function x p = M X k = 1 c k x p k solves the nonhomogeneous linear system L [ y ] = M X k = 1 c k f k . (This follows immediately from the fact that L is a linear operator.) Hand Examples In our first example, we’ll do things soup-to-nuts by hand. In the next example, we’ll assume we have the necessary eigenpairs (i.e., pairs of eigenvalues with associated eigenvectors) so as to rapidly form a general solution of the associated homogeneous system. Then we’ll proceed to the main course: finding a particular solution of the nonhomogeneous system. 555/2 Find a general solution to the nonhomogeneous system x 0 = Ax + f ,where A = ± 11 41 ² and f = ± - t - 1 - 4 t - 2 ² . Solution Here is our overall solution strategy. 1. Find a general solution x h to the associated homogeneous system x 0 = Ax . 2. Find a particular solution x p to the nonhomogeneous system x 0 = Ax + f . 3. Form a general solution of the nonhomogeneous system: x = x p + x h . Along the way, we’ll flesh out details. Let’s get the party started. 1. Computation of x h . (a) Eigenvalues of A. Solve det ( A - r I ) = 0. ³ ³ ³ ³ 1 - r 1 - r ³ ³ ³ ³ = r 2 - 2 r - 3 = ( r + 1 )( r - 3 ) = 0, whence r =- 1 , 3. (b) Associated Eigenvectors. Find a nonzero vector in the nullspace of the RREF of A - r I .
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This note was uploaded on 03/29/2008 for the course MATH 308 taught by Professor Comech during the Spring '08 term at Texas A&M.

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L97M - Fall 2003 Math 308/501502 9 Matrix Methods for...

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