Answer 1

`dy/dx - y.cosecx = tan x/2`

`dy/dx - y.cosecx = tan x/2`

`dy/dx - y.cosec x = tan x/2`   ....(i)

Compare `dy/dx + Py = Q`

P = -cosecx, Q = `tan  x/2`

`I.F. = e^(intPdx)`

`I.F. = e^(int-cosecx dx)`

`I.F. = e^(-log_e(cosecx- cotx))`

`I.F. = e^(log_e(cosecx - cot x)^(-1))`

`I.F. = (cosec x - cot x)^(-1) =  (cosect x + cot x)`

∴ solution of the linear differential equation

`y.I.F. = int Q.IF.dx`

`y.(cosecx + cotx)  = int tan  x/2 ((1+cosx)/sinx) dx`

`y.(cosecx - cot x) = int (sin  x/2)/(cos  x/2) . (2cos^2  x/2)/(2sin  x/2 .cos  x/2) dx = int 1 dx`

y.(cosec x + cot x) = x + c