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282_pdfsam_math 54 differential equation solutions odd

282_pdfsam_math 54 differential equation solutions odd -...

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Chapter 5 By simplifying these equations, we observe that this cooling problem satisfies the system 4 x ( t ) + 3 x ( t ) y ( t ) = 200 , x ( t ) + 4 y ( t ) + 2 y ( t ) = 52 . In operator notation, this system becomes (4 D + 3)[ x ] [ y ] = 200 , [ x ] + (4 D + 2)[ y ] = 52 . 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 first 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 + 20 r + 5 = 0. By the quadratic
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