This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 firstorder 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 lefthandside: 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 rewrite 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 firstorder equations, and ended up with a secondorder 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 vectorvalued function....
View
Full
Document
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.
 Spring '10
 None

Click to edit the document details