Give the correct answer and solve the given equation dx+(xy−sin⁡y)dy=0

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Give the correct answer and solve the given equation dx+(xy​−sin⁡y)dy=0

brawnyN · Accepted Answer

Multiply this equation by y:ydx+(x​−ysin⁡y)dy=0Now we will try to find a function F: RR2→R such that∂F∂x=yand ∂F∂x=x−ysin⁡y (1)Start with ∂F∂x=yand integrate with respect to x:F(x,y)=∫ydx=xy+C(y)(2)where C(y) is a constant with respect to x (an integrating constant!). From (2)∂F∂x=x+C′(y)Using (1) now, we get thatx+C′(y)=x−ysin⁡y⇒C′(y)=−ysin⁡yThus,C(y)=∫−ysin⁡ydy=−∫ysin⁡ydy=(u=y⇒du=dy),(dv=sin⁡ y dy⇒v=−cos⁡y)=−(y(−cos⁡ y)−∫−cos⁡ y dy)=ycos⁡y−∫cos⁡ydy=ycos⁡y−∼y+D,where D is a constant.Therefore,F(x,y)=xy+C(y)=xy+ycos⁡y−sin⁡y+DNow the solution of the initial differential equation is given by F(x,y)=0,so xy+ycos⁡y−sin⁡y=−D=C,where C is some constant.

Give the correct answer and solve the given equation dx + (frac{x}{y} − sin y)dy = 0

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