Solve the following differential equation: `cos^2\ x(dy)/(dx)+y=tan\ x`

Explanation

`cos^(2)x(dy)/(dx)+y=tanx`
` (dy)/(dx) + y/(cos^(2)x) = tanx/(cos^(2)x)`
` (dy)/(dx) +sec^(2)x.y =sec^(2)x . tan x`
This is the linear differential equation of the form.
`(dy)/(dx) + P.y =Q, " where" P = sec^(2)x and Q =sec^(2) x . tanx`
` 1.F. e^(int sec^(2)xdx) =e^(tanx)`
The solution of (1) is given by
`y.(I.F.) = int Q . (I.F.) dx +c `
`y.e^(tanx) = int sec^(2) .tan x .e^(tan x) dx +c`
` e^(tanx).y = int e^(tanx ) sec^(2) . tan x dx +c`
Put ` tan x =t " " therefore sec^(2) xdx =dt`
`e^(tanx .) y= int e^(t) . tdt +c`
` therefore e^(tanx) .y =t int e^(t) .dt - int [d/(dt)(t) e^(t) dt] dt +c`
` e^(tan x) .y = t e^(x) - int 1.e^(t) dt +c`
` e^(tan x.) y = t e^(t) -e^(t) +c " " therefore e^(tanx ) .y =e^(t) (t-1) +c`
` e^(tan x)/y =e^(tan x) (tan x-1) +c " " therefore y= (tan x-1) +c.e^(-tanx)`
This is the general solution .