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Question
For the differential equation, find the general solution:
`dy/dx + (sec x) y = tan x (0 <= x < pi/2)`
Solution
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
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