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Unformatted text preview: Solving Non-homogeneous Systems of Equations These notes describe how to solve (i.e., find the general solution of) a differential equation of the form d x dt = A ( t ) x + g ( t ) , when A is a 2 2 matrix. Note carefully that we assume that the d x dt term has coefficient 1. There are three methods: #1: Change of basis. #2: Undetermined coefficients. #3: Variation of parameters. Methods #1 and #2 can only be used if A ( t ) is independent of t , i.e., a constant matrix. Moreover, method #2 requires g to be of specific form: it can only contains terms that involve positive powers of t , e at , cos( at ), and sin( at ). Method #3 can always be used. It is recommended that you use method #3 because it always applies and is usually shorter than method #1. When method #2 can be used, it involves easier math (if you find solving vector equations easier than integration) but it can be tricky to set up for certain equations. Method #1: change of basis The idea here is to find a matrix P such that P- 1 AP is of a certain form and then introducing a new variable y = P- 1 x . The equation for x leads to an equation for y , namely d y dt = P- 1 AP y + P- 1 g ( t ) . This equation will be easier to solve than the original and once you solve for y , you then simply use the fact that x = P y to get the solution for x . There are three cases depending on A (for details see the notes on exponentials of ma- trices)....
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