Question

For the differential equation, find the general solution:

`dy/dx + (sec x) y = tan x (0 <= x < pi/2)`

Answer 1

This is a linear differential equation of the form `dy/dx  Py = Q`

where P = sec x and Q = tan x

∴ I.F. = `e^(int P dx) = e^(int sec x  dx)`

`= e^(log (sec x + tan x))` = (sec x + tan x)

Hence, the solution of the differential equation

∴ `y xx I.F. = int Q xx I.F. dx + C`

⇒  `y(sec x + tan x) = int tan x xx (sec x + tan x)dx + C`

⇒ `y(sec x + tan x) = int (tan sec x + tan^2 x) dx + C`

`⇒ y (sec x + tan x) = int tan sec x  dx + int sec^2 x  dx -  int 1  dx + C`

⇒ y(sec x + tan x) = sec x + tan x - x + C