Question

Find the general solution: $\frac {dy}{dx}+(sec\ x)y=tan x, \ (0≤x<\frac \pi2)$

Answer 1

The given differential equation is:

$\frac {dy}{dx}$+py = Q(where p = sec x and Q = tan x)

Now, I.F = e∫pdx = $\int$sec x dx = elog(sec x+tan x) = sec x+tan x

The general solution of the given differential equation is given by the relation,

y(I.F.) = $\int$(Q×I.F.)dx + C

⇒y(secx + tanx) = $\int$tan x(sec x + tan x)dx +C

⇒y(sec x + tan x) =$\int$sec x tan x dx + $\int$tan2x dx + C

⇒y(sec x + tan x) = sec x +$\int$(sec2x - 1)dx + C

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