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hw_2_soln_corrected

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 ± dy dx =− 3 2 x + 3 + ° 2 y 2 ± = 0 (2 x + 3) + ° 2 y 2 ± = 0 This differential is exact by the exactness criterion: y (2 x + 3) = 0; x ° 2 y 2 ± = 0. We want f such that df ° x , y ± = (2 x + 3) + ° 2 y 2 ± , 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 ² x 2 3 x + C y = 1 ± ² x 2 3 x + C 1
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2. Rewrite in differential form: dy dx = 5 x + 4 y 8 y 3 4 x ° 8 y 3 4 x ± = 5 x + 4 y ° 5 x 4 y ± + ° 8 y 3 4 x ± = 0 ° 5 x 4 y ± + ° 8 y 3 4 x ± = 0 This differential is exact: y ° 5 x 4 y ± =− 4; x ° 8 y 3 4 x ± 4.
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hw_2_soln_corrected - [Name Section Homework#2 Math 527 1...

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