lecture_41 - MA 265 LECTURE NOTES: WEDNESDAY, APRIL 23...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MA 265 LECTURE NOTES: WEDNESDAY, APRIL 23 Systems with Diagonalizable Matrices Example. Consider the first-order homogeneous system dx dt = x ( t )- y ( t ) dy dt = 2 x ( t ) + 4 y ( t ) We find the general solution. The idea is to make a change of variables to make things simpler. First, we compute the eigenvalues and eigenvectors of the coefficient matrix. Lets write this system in the form d x dt = A x ( t ) where A = 1- 1 2 4 and x ( t ) = x ( t ) y ( t ) . The characteristic polynomial of A is p ( ) = det[ I 2- A ] = - 1 1- 2 - 4 = 2- 5 + 6 = ( - 2)( - 3) so that the eigenvalues are 1 = 2 and 2 = 3. We see that the eigenvectors are A p 1 = 1 p 1 for p 1 = 1- 1 ; A p 2 = 2 p 2 for p 2 = 1- 2 . Define the following 2 2 matrix using these eigenvectors: P = 1 1- 1- 2 = D = P- 1 AP = 2 1- 1- 1 1- 1 2 4 1 1- 1- 2 = 2 0 0 3 . Second, we make a change of variables to find a simpler first-order homogeneous system. Denote u ( t ) = P- 1 x ( t ) = u ( t ) = 2 x ( t ) + y ( t ) v ( t ) =- x ( t )- y ( t ) We have the following differential equations: du dt = 2 dx dt + dy dt = 2 x ( t )- y ( t ) + 2 x ( t ) + 4 y ( t ) = 4 x ( t ) + 2 y ( t ) = 2 u ( t ) dv dt =- dx dt- dy dt =- x ( t )- y ( t )- 2 x ( t ) + 4 y ( t ) =- 3 x ( t )- 3 y ( t ) = 3 v ( t ) =...
View Full Document

This note was uploaded on 03/11/2010 for the course MA 261A 0026100 taught by Professor ... during the Spring '10 term at Purdue University Calumet.

Page1 / 3

lecture_41 - MA 265 LECTURE NOTES: WEDNESDAY, APRIL 23...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online