The solution of (dy)/(dx)+x sin 2y=x^(3) cos^(2) y, is

Explanation

We have on dividing by `cos^(2)y`
`sec^(2)y(dy)/(dx) + ((2sin y cos y)/(cos^(2) y))x = x^(3)`
`rArr sec^(2)y(dy)/(dx) + (2 tan y) x = x^(3)`
Set `tan y = v`, so that `sec^(2) y (dy)/(dx) = (dv)/(dx)`
Our equation becomes,
`(dv)/(dx) + 2vx = x^(3)`
which is linear in v
`IF = e^(int 2x dx) = e^(x^(2))`
multiplying the equation by If and integrating
`ve^(x^(2)) = int e^(x^(2)). x^(3)dx + k`, k n being the constant of integration
`= int e^(x^(2)). x^(2).xdx + k` (set `t = x^(2)`)
`= (1)/(2) int e^(t). t dt + k`
`= (1)/(2)e^(t).(t-1) + k`
`rArr - tan y e^(x^(2)) = (1)/(2)e^(x^(2)) (x^(2)-1) + k`
`rArr 2 tan y e^(x^(2)) (x^(2)+1) + lambda, lambda` being another constant.
`:. 2 tan y = (x^(2) - 1) + lambda e^(-x^(2))`.