You found the homogeneous solution to be a combination of terms involving sin(t) and cos(t). Since your inhomogeneous term (-cos(t), sin(t)) is of the same functional form, the inhomogeneous solution can contain terms of the form
x= (A1, A2) cos(t) + (B1, B2) sin(t) + (C1, C2) t cos(t) + (D1, D2) t sin(t)
To determine the 8 unknown coefficients A1 through D2 you would substitute this x into your original equation and get 8 linear equations.
An easier method in this case is given by variation of parameters.
Andre W.
tutor
The method was explained by Kirill here: http://www.wyzant.com/resources/answers/14707/find_the_general_solution
There is a shortcut for this method and it goes like this: Find a fundamental matrix X for the homogeneous system and find its inverse X-1. Then if b is the inhomogeneous term, the inhomogeneous part of the solution is given by
x = X ∫ X-1b dt.
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10/02/13
Sun K.
What original equation do I plug this x into it to find 8 different values? And how do you get the 8 linear equations?
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10/02/13
Andre W.
tutor
You plug x into x'=(2, 1, -5, -2)x+(-cos(t), sin(t)). On the left side, you need to take the derivative. On the right side, you need to do matrix multiplication. It will look messy.
Since this is a vector equation, you really have 2 equations, one for the first component and one for the second component. Write them out. From each of these two equations, you extract 4 equations by comparing the terms with a cos(t), with a sin(t), with a t cos(t), and with a t sin(t) on the left and right sides. You get a total of 8 equations.
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10/03/13
Sun K.
10/02/13