284_pdfsam_math 54 differential equation solutions odd

284_pdfsam_math 54 differential equation solutions odd -...

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Chapter 5 Room B. Similarly, we obtain y 0 = ± 1000 · 2 1000 + 1 2 ( x y ) ² 1 5 ( y 0) = 2 + 1 2 x 7 10 y. Hence, the system governing the temperature exchange is x 0 =20 (3 / 4) x +(1 / 2) y, y 0 =2+(1 / 2) x (7 / 10) We fnd the critical points oF this system by solving 20 (3 / 4) x / 2) y =0 , 2+(1 / 2) x (7 / 10) y 3 x 2 y =8 0 , 5 x +7 y =2 0 x = 600 / 11 , y = 460 / 11 . ThereFore, (600 / 11 , 460 / 11) is the only critical point oF the system. Analyzing the direction feld, we conclude that (600 / 11 , 460 / 11) is an asymptotically stable node. Hence, lim t →∞ y ( t )= 460 11 41 . 8 ± . (One can also fnd an explicit solution y ( t ) = 460 / 11 + c 1 e r 1 t + c 2 e r 2 t ,where r 1 < 0, r 2 < 0, to conclude that y ( t ) 460 / 11 as t →∞ .) 39. Let y be an arbitrary Function di²erentiable as many times as necessary. Note that, For a di²erential operator, say, A , A [ y
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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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