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hw_2_soln_corrected - [Name Section Homework#2 Math 527 1...

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[Name] Section [#] Homework #2 Math 527 1. Rewrite in differential form: 2 x + 2 y 2 d y dx = 3 2 x + 3 + 2 y 2 d y dx = 0 (2 x + 3) dx + 2 y 2 d y = 0 This differential is exact by the exactness criterion: y (2 x + 3) = 0; x 2 y 2 = 0. We want f such that d f x , y = (2 x + 3) dx + 2 y 2 d y , that is f x x , y = 2 x + 3 and f y x , y = 2 y 2 f x , y = x 2 + 3 x + g y . . . f y x , y = g y and f y x , y = 2 y 2 (1) Combine the last pair of equations: g y = 2 y 2 g y = y 2 2 y (2) Substitute (2) into equation (1): f x , y = x 2 + 3 x + y 2 2 y The general solution is x 2 + 3 x + y 2 2 y = C ...which can be made explicit: y 2 2 y = x 2 3 x + C y 2 2 y + 1 = x 2 3 x + C ( C : = C + 1) y 1 2 = C x 2 3 x + C y = 1 = ± x 2 3 x + C y = 1 ± x 2 3 x + C 1
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2. Rewrite in differential form: d y dx = 5 x + 4 y 8 y 3 4 x 8 y 3 4 x d y dx = 5 x + 4 y 5 x 4 y + 8 y 3 4 x d y dx = 0 5 x 4 y dx + 8 y 3 4 x d y = 0 This differential is exact: y 5 x 4 y = 4; x 8 y 3 4 x = 4.
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