Question

Solve the following differential equation:

`"dy"/"dx" + "y" * sec "x" = tan "x"`

Answer 1

`"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.