I have tried to make this equation seperable, homogeneous, exact and use integrating factors and nothing seems to be working. Could you tell me what form to use?
Find the general solution of $(x + \sin{x} +\sin{y})\ \textrm{d}{x} + \cos{y}\ \textrm{d}{y}=0$
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ordinary-differential-equations
1 Answers
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if you multply through by $e^x$ you have an exact diff eq.
$(x e^x + e^x \sin x + e^x \sin y)dx + (e^x \cos y) dy = 0\\ \nabla (x e^x-e^x + \frac 12 e^x \sin x - \frac 12 e^x \cos x + e^x \sin y) = (x e^x + e^x \sin x + e^x \sin y)dx + (e^x \cos y) dy\\ x e^x-e^x + \frac 12 e^x \sin x - \frac 12 e^x \cos x + e^x \sin y = C\\ -x + 1 - \frac 12 \sin x + \frac 12 \cos x +Ce^{-x} = \sin y\\ y = \arcsin( 1-x - \frac 12 \sin x + \frac 12 \cos x +Ce^{-x})$
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0How did you find e^x as the integrating factor? – 2017-01-27
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0We need a factor such that $\frac {d}{dy} f(x) \sin y = \frac {d}{dx} f(x) \cos y$ – 2017-01-27