Integrating factor/solve differential equation

Question

I am asked to find the integrating factor and solve.

$$ y\sin(y)dx + x(\sin(y) - y\cos(y))dy = 0.$$

I'm not sure on how to put this in the form of

$$y' + p(x)y = f(x)$$

to solve the equation. Or is there another method to use?

Answer 1

You can separate dx and dy here:

$\frac {-(sin(y)−ycos(y))}{ysin(y)}dy=\frac {dx}{x}$

And solve the now separated variables equation, can you go on from here?

Answer 2

For an equation in the form $$Mdx +Ndy=0$$ where $M$ and $N$ are functions of $x$ and $y$ the integrating factor is $$\mu = e^{\displaystyle\int (M_y - N_x)/N dx}$$