Solve: `(xcosy-ysiny)dy+(x siny+ycosy)dx=0`

Explanation

We have
`(xcosy-ysiny)(dy)/(dx)+(xsiny+ycosy)=0`.............(1)
Let `xsiny+ycosy=t`
Differentiating w.r.t. `x`, we get
`xcosy(dy)/(dx)+siny+cosy(dy)/(dx)-ysiny(dy)/(dx)=(dt)/(dx)`
`therefore (xcosy-ysiny)(dy)/(dx)+siny+cosy(dy)/(dx)=(dt)/(dx)`
So, from (1), we have
`(dt)/(dx)-(siny+(d(siny))/(dx))+t=0`
`rArr (d(t-siny))/(dx) = -(tsiny)`
`rArr (d(t-siny))/(t-siny)=-dx`
`rArr int(d(t-siny))/(t-siny) =-intdx`
`rArr log_(e)(t-siny)=-x+log_(e)C`
`rArr t-siny=Ce^(-x)`
`rArr xsiny+ycosy-siny=Ce^(-x)`