282_pdfsam_math 54 differential equation solutions odd

282_pdfsam_math 54 - Chapter 5 By simplifying these equations we observe that this cooling problem satises the system 4x(t 3x(t y(t = 200 x(t 4y(t

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Chapter 5 By simplifying these equations, we observe that this cooling problem satisFes the system 4 x 0 ( t )+3 x ( t ) y ( t ) = 200 , x 0 ( t )+4 y 0 ( t )+2 y ( t )=5 2 . In operator notation, this system becomes (4 D +3)[ x ] [ y ] = 200 , [ x ]+(4 D +2)[ y ]=5 2 . Since we are interested in the temperature in the attic, x ( t ), we will eliminate the function y ( t ) from the system above by applying (4 D +2) to the Frst equation and adding the resulting equations to obtain { (4 D + 2)(4 D +3) 1 } [ x ]=(4 D + 2)[200] + 52 = 452 ( 16 D 2 +20 D +5 ) [ x ] = 452 . (5.23) This last equation is a linear equation with constant coefficients whose corresponding homo- geneous equation has the associated auxiliary equation 16 r 2 r + 5 = 0. By the quadratic formula, the roots to this auxiliary equation are
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

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