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Question Number 169913 by ali009 last updated on 12/May/22
solve the D.E  dx+(−sin(y)+(x/y))dy=0
$${solve}\:{the}\:{D}.{E} \\ $$$${dx}+\left(−{sin}\left({y}\right)+\frac{{x}}{{y}}\right){dy}=\mathrm{0} \\ $$
Commented by cortano1 last updated on 12/May/22
 sin y dy −(x/y) dy = dx    y sin y dy = x dy +y dx    y sin y dy = d(xy)   ∫ y sin y dy = ∫ d(xy)   sin y− y cos y = xy + c
$$\:\mathrm{sin}\:{y}\:{dy}\:−\frac{{x}}{{y}}\:{dy}\:=\:{dx}\: \\ $$$$\:{y}\:\mathrm{sin}\:{y}\:{dy}\:=\:{x}\:{dy}\:+{y}\:{dx}\: \\ $$$$\:{y}\:\mathrm{sin}\:{y}\:{dy}\:=\:{d}\left({xy}\right) \\ $$$$\:\int\:{y}\:\mathrm{sin}\:{y}\:{dy}\:=\:\int\:{d}\left({xy}\right) \\ $$$$\:\mathrm{sin}\:{y}−\:{y}\:\mathrm{cos}\:{y}\:=\:{xy}\:+\:{c} \\ $$

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