Two-DimensionalSystems

# Two-DimensionalSystems - ) = --+ 1 2 2 2 2 2 which reduces...

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TWO-DIMENSIONAL SYSTEMS Method for solving a two-dimensional first order system of differential equations from section 4.1 p228. 13. p228 x y y x ' ' = - = 2 2 x y ( ) ( ) 0 1 0 0 = = Given a system of differential equations and initial conditions, find the general solutions and the particular solutions. Take the derivative of the first expression x y " ' = - 2 Substitute 2 x for y ' x x " ( ) = - 2 2 Now solve the resulting second order equation x x " + = 4 0 The characteristic equation is r 2 4 0 + = which has complex roots r i = ± 0 2 This gives us the general solution x t e A t B t t ( ) ( cos sin ) = + 0 2 2 Taking the derivative gives x t A t B t '( ) sin cos = - + 2 2 2 2 The first problem expression can be rewritten y x = - ' 2 so that y t x t ( ) '( ) = - 1 2 by substituting for x '( t ) we have y t A t B t ( ) ( sin cos
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Unformatted text preview: ) = --+ 1 2 2 2 2 2 which reduces to y t A t B t ( ) sin cos = + 2 2 we now have this general solution to the system of equations to which we can apply the initial conditions x t A t B t ( ) ( cos sin ) = + 2 2 y t A t B t ( ) sin cos = + 2 2 General solutions: x t A t B t ( ) cos sin = + 2 2 y t A t B t ( ) sin cos = + 2 2 Initial conditions: x ( ) 1 = y ( ) = substituting: 1 = + A B cos sin = + A B sin cos determines the constants: A = 1 B = This yields the particular solutions: x t t ( ) cos = 2 y t t ( ) sin = 2 Tom Penick [email protected] October 24, 1997...
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## This note was uploaded on 03/29/2008 for the course MATH 308 taught by Professor Comech during the Spring '08 term at Texas A&M.

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