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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 = 1000 · 2 1000 + 1 2 ( x y ) 1 5 ( y 0) = 2 + 1 2 x 7 10 y. Hence, the system governing the temperature exchange is x = 20 (3 / 4) x + (1 / 2) y, y = 2 + (1 / 2) x (7 / 10) y. We find the critical points of this system by solving 20 (3 / 4) x + (1 / 2) y = 0 , 2 + (1 / 2) x (7 / 10) y = 0 3 x 2 y = 80 , 5 x + 7 y = 20 x = 600 / 11 , y = 460 / 11 . Therefore, (600 / 11 , 460 / 11) is the only critical point of the system. Analyzing the direction field, we conclude that (600 / 11 , 460 / 11) is an asymptotically stable node. Hence, lim t →∞ y ( t ) = 460 11 41 . 8 F . (One can also find 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 differentiable as many times as necessary. Note that, for a differential operator, say,
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