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Question
Solve the following differential equation:
`"dy"/"dx" + "y" * sec "x" = tan "x"`
Solution
`"dy"/"dx" + "y" * sec "x" = tan "x"`
∴ `"dy"/"dx" + (sec "x") * "y" = tan "x"` .....(1)
This is the linear differential equation of the form
`"dy"/"dx" + "P" * "y" = "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
∴ the solution of (1) is given by
∴ y(I.F.) = `int "Q" * ("I.F.") "dx" + "c"`
∴ y (sec x + tan x) = ∫ tan x (sec x + tan x) dx + c
∴ (sec x + tan x) y = ∫ (sec x tan x + tan2x) dx + c
∴ (sec x + tan x) y = ∫ (sec x tan x + sec2x - 1) dx + c
∴ (sec x + tan x) y = sec x + tan x - x + c
∴ y (sec x + tan x) = sec x + tan x - x + c
This is the general solution.
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