Show that the differential equation `(y sin^(2)((x)/(y))-x)dy+ydx = 0` is homogeneous differential equation. Also find its general solution.

Explanation

`(y sin^(2)((x)/(y))-x)dy+ydx = 0`
`rArr (dx)/(dy) = (x-y sin^(2) ((x)/(y)))/(y)`
Let `f(x,y) = (x-y sin^(2)((x)/(y)))/(y)`
`rArr f(lambda x, lambda y) = (lambda x - lambda y sin^(2)((lambda x)/(lambda y)))/(lambda y)`
`= lambda^(0)((x-y sin^(2)((x)/(y)))/(y))`
The given differential equation is the homogeneous differential equation.
Putting x = vy
`rArr (dx)/(dy) = v + y(dv)/(dy)` ...(1)
We get,
`v+y(dv)/(dy) = (vy -y sin^(2)((vy)/(y)))/(y)`
`rArr y(dv)/(dy) = -sin^(2)v`
`rArr cosec^(2) vdv = -(dy)/(y)`
Integrating both sides, we get
`-cot v = In y - In c`
`rArr cot ((x)/(y)) = In|cy|`