systems_intro_real - System of Eq. Distinct and Real...

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Unformatted text preview: System of Eq. Distinct and Real eigenvalues April 16, 2011 1 Introduction/Motivation Suppose were given the follow system of differential equations and asked to solve it: Note: here x is the dependent variable, and t is the indepen- dent variable x 1 ( t ) = 1 x 1 ( t ) + 1 x 2 ( t ) (1) x 2 ( t ) = 4 x 1 ( t ) + 1 x 2 ( t ) (2) If x 2 ( t ) did not appear in the first equation, wed have a simple first-order equation, and could certainly solve it. Therefore, it is only reasonable to try and eliminate x 2 ( t ) from Eq. ( 2 ). Using Eq. ( 1 ), we find x 2 ( t ) = x 1 ( t )- x 1 ( t ) . (3) Plugging this expression into Eq. ( 2 ) we have ( x 1 ( t )- x 1 ( t )) = 4 x 1 ( t ) + 1 ( x 1 ( t )- x 1 ( t )) . Expanding the left-hand-side: x 00 1 ( t )- x 1 ( t ) = 4 x 1 ( t ) + x 1 ( t )- x 1 ( t ) , or finally, x 00 1- x 1- 3 x 1 = 0 . (4) We can easily find the solution to above equation. The answer ends up being x 1 ( t ) = c 1 e 3 t + c 2 e- t . Using above expression and Eq. ( 3 ), we can easily solve for x 2 ( t ) . The result turns out to ! We wont absorb the 2s into the c 1 and c 2 here. be x 2 ( t ) = 2 c 1 e 3 t- 2 c 2 e- t . (5) To make a few observations, we will re-write our results in the following vector form: x 1 x 2 = c 1 e 3 t 2 c 1 e 3 t + c 2 e- t- 2 c 2 e- t = c 1 e 3 t 2 e 3 t + c 2 e- t- 2 e- t = c 1 1 2 e 3 t + c 2 1- 2 e- t Key observations: 1 c HF, 2011 System of Eq. Distinct and Real eigenvalues April 16, 2011 1. We started with a system of first-order equations, and ended up with a second-order equation. 2. Our solution is in the following form: x 1 x 2 = 1 2 e rt , where 1 and 2 and constants. In a more compact form, above equation may be written as x ( t ) = e rt , (6) where x (t) (bold) denotes a vector-valued function....
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This note was uploaded on 05/02/2011 for the course GE 207K taught by Professor None during the Spring '10 term at University of Texas at Austin.

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systems_intro_real - System of Eq. Distinct and Real...

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