This preview shows pages 1–2. 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: MA366 Sathaye Fast methods for some chapter 7 problems. We show some faster methods to quickly get a fundamental matrix for linear systems of differential equations. Two variable systems. 1. Basic formulas. We consider a system X = AX where A is a 2 2 matrix with constant coefficients. Let A = a b c d . We note that its characteristic polynomial has the formula P A ( ) = 2 ( a + d ) + ad bc . We typically solve for eigenvalues by finding the roots of P A ( ) and depending on the nature of roots, we have different techniques for solving the system. 2. Fundamental Matrix For this discussion, we may even consider an n n matrix A giving a system of n equations in n variables. A Fundamental matrix for the system X = AX is defined to be an n n matrix ( t ) whose columns are n independent solutions of the system. This can be shown equivalent to the condition ( t ) = A ( t ) together with the condition that ( t ) is invertible on an interval where the solutions are discussed. The Special Fundamental Matrix is given by ( t ) = exp ( At ) = I + At + A 2 t 2 / 2! + Its columns are independent solutions of the system and have the added property that (0) = I . Indeed, this extra condition identifies a Fundamental Matrix as a Special Fundamental Matrix. A general solution to the system X = AX is then given by X ( t ) = ( t ) C where C is a column of arbitrary constants. In particular, X = ( t ) X (0) where X (0) = X (0) is the given set of initial conditions. We give direct methods to find this matrix ( t ). 3. Exponential shift. Consider a substitution X = exp( rt ) Y . Substitution in the equation X = AX gives X = exp( rt ) rY + exp( rt ) Y = A exp( rt ) Y. By canceling exp( rt ) and rearranging, we see that Y = ( A rI ) Y. It is easy to see that the characteristic polynomial of A rI is simply P A ( + r ). We shall use this substitution to simplify our characteristic polynomial. Finally, whatever special fundamental matrix is found for the system Y = ( a rI ) Y , we may multiply it by exp( rt ) to get the matrix for the system X = AX ....
View
Full
Document
This note was uploaded on 01/18/2012 for the course MATH 366 taught by Professor Edraygoins during the Fall '09 term at Purdue UniversityWest Lafayette.
 Fall '09
 EdrayGoins
 Differential Equations, Equations, Linear Systems, Formulas

Click to edit the document details