Solve the differential equation?

Question

Solve the differential equation dx+(x/y-sin(y))dy=0.

I know that the integrating factor is y.

y dx+x dy-ysin(y) dy=0

d(xy)-ysin(y) dy=0

How do I integrate this?

Answer 1

Separate variables,

d(xy) = ysin(y) dy

Integrate both sides,

xy = ∫ysin(y) dy = -ycos(y) + sin(y) + c <==Answer

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Attn:

∫ysin(y) dy = -ycos(y) + sin(y) + c can be obtained by tabler method or integration by parts.