hw_2_soln_corrected

Which will be left implicit 4 rewrite in differential

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Unformatted text preview: y sin x d x + e x cos y + 2 cos x d y , = = = e x sin y + 2 y sin x ￿ ￿ f y x, y and ￿￿ e x sin y − 2 y cos x + g ￿ y￿ e x cos y − 2 cos x + g ￿ y ￿ ￿ fx x, y and Combine the last pair of equations: ￿￿ e x cos y − 2 cos x + g ￿ y ￿￿ g￿ y ￿￿ gy = . . . = = e x cos y + 2 cos x (5) e x cos y + 2 cos x = e x cos y + 2 cos x 0 0 = (6) Substitute (6) into (5): ￿ ￿ f x , y = e x sin y − 2 y cos x The general solution is e x sin y − 2 y cos x = C ...which will be left implicit. 4. Rewrite in differential form: ￿ ￿ dy x ln y + x y + y ln x + x y ￿ ￿ ￿ ￿d x x ln y + x y d x + y ln x + x y d y = 0 = 0 This differential is not exact: ￿ ∂￿ x ln y + x y d x ∂y ￿ ∂￿ y ln x + x y ∂x = = x + x; y y + y. x (These two are not equal, for example, when x = 1 and y = 2.) 5. Rewrite in differential form: ￿ ￿ dy x − y 3 + y 2 sin x + −3x y 2 − 2 y cos x dx ￿ ￿ ￿ ￿ 3 2 2 x − y + y sin x d x + −3x y − 2 y cos x d y = 0 = 0 This differential is exact: ￿ ∂￿ x − y 3 + y 2 sin x ∂y ￿ ∂￿ −3x y 2 − 2 y cos x ∂x 3 =...
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