Show that the differential equation `x sin ((y)/(x))(dy)/(dx) = y sin ((y)/(x)) + x` is homogeneous. Also find its general solution.

Explanation

`(dy)/(dx) = (y sin((y)/(x))+x)/(x sin ((y)/(x)))` ...(1)
Let `f(x, y) = (y sin ((y)/(x))+x)/(x sin ((y)/(x)))`
Now `f(lambda x, lambda y) = (lambda y sin((lambda y)/(lambda x)) + lambda x)/(lambda x sin ((lambda y)/(lambda x)))`
`= lambda_(0)((ysin ((y)/(x)) + x)/(x sin((y)/(x))))`
`= lambda^(0)f(x, f)`
Thus the given differential equation is homogeneous.
Now putting y = vx
`rArr (dy)/(dx) = v+x(dv)/(dx)` in (1), we get
`v + x(dv)/(dx) = (vx sin v + x)/(x sin v)`
`rArr x(dv)/(dx) = (v sin v + 1 - v sin v)/(sin v)`
`sin v dv = (dx)/(x)`
Integrating both sides, we get - cos v = In x + Inc
`rArr - cos.(y)/(x) = In |c x|`
Which is the general solution of the differential equation (1)