Question

Solve the differential equation `cos^2 x dy/dx` + y = tan x

Answer 1

`cos^2 x dy/dx` + y = tan x

∴ `dy/dx + y/(cos^2x) = tanx/(cos^2x)`

∴ `dy/dx + sec^2x.y` = tan x . sec² x

The given equation is of the form

`dy/dx + Py` = Q,

Where P = sec² x and Q = tan x. sec² x

∴ I.F. = `e^(int Pdx) = e^(intsec^2x  dx)` = e^{tan x}

∴ Solution of the given equation is

y(I.F.) = `int Q.(I.F.)dx + c`

∴ ye^{tan x} = `int tan x.sec^2x.e^(tanx)dx+ c`

Put tan x = t

∴  sec²x dx = dt

∴  ye^{tan x} = `int te^t dt + c`

= `tint e^t dt - int[d/dt (t) inte^tdt]dt + c`

= `te^t - int e^tdt + c`

= te^t – e^t + c

∴ ye^{tan x} = e^tanx (tanx – 1) + c

∴ y = tan x – 1 + c.e^–tanx