Find the general solution of the differential equations:`(dx)/(dy)+secx y=tanx(0lt=xltpi/2)`

Options

A. ` y (sec x - tanx) = (sec x + tan x) -x +C`
B. ` y (sec x+tanx)=(sec x - tan x)-x+C`
C. ` y (sec x + tan x) = (sec x + tan x)-x+C`
D. None of the above

Answer

c

Explanation

The given differential equation is ` (dy)/(dx) + y sec x = tan x ` ,
On comparing with the form `(dy)/(dx) +PY = Q`
` P = sec x and Q = tan x`
` :. IF = e^(int Pdx) = e^(int sec xdx)`
` rArr IF = e^(log|sec x + tan x | )= sec x + tan x" "` ...(i)
The general solution of the differential equation is given by
` y * IF = int Q xx IF dx + C`
` rArr y (sec x + tan x) = int tan x (sec x +tan x) dx+C`
` rArr y ( sec x + tan x) = int tan x sec x dx + int tan^(2) x dx +C`
` rArr y ( sec x +tan x) = sec x + int sec^(2) x dx - int 1 dx +C`
` [ :' tan ^(2) x = sec ^(2) x - 1]`
` rArrr y (sec x + tan x) = sec x + tan x - x +C`