Solve the differential equation `cos^(2) ydx + xdy = tan ydy`.

Explanation

The given differential equation can be written as
`(dx)/(dy) + (x)/(cos^(2)y) = (tan y)/(cos^(2)y)` ...(1)
Which is a linar differential equation of the type `(dx)/(dy) + Px = Q`, where `P = (1)/(cos^(2)y)` and `Q = (tan y)/(cos^(2) y)`
`:. I.F. = e^(int sec^(2) ydy) = e^(tan y)`
Thus, the solution of (1) is
`xe^(tan y) = int e^(tan y). tan y. sec^(2)ydy + c`
Putting tany = t in the integral
`rArr xe^(tan y) = int e^(t). tdt + c = te^(t) - int 1.e^(t) dt + c = e^(t)(t-1) + c = e^(tan y)(tan y - 1) + c`
`rArr x = (tan y - 1) + ce^(-tan y)`