Question

For the differential equation given, find a particular solution satisfying the given condition:

`dy/dx + 2y tan x = sin x; y = 0 " when x " = pi/3`

Answer 1

The given equation is

`dy/dx + 2y tan x = sin x`

Which is a linear equation of the type

`dy/dx + Py = Q`

Hence P = 2 tan x and Q = sin x

∴ `int Pdx = int 2 tan x dx = 2 log |sec x| = log sec^2 x`

∴ `I.F. = e^(int Pdx) = e^(log sec^2x) = sec^2 x`

∴ The solution is `y. (I.F.) = int Q. (I.F.)  dx  + C`

⇒ `y sec^2 x = int sin x sec^2 x  dx + C`

`= int sec x tan x  dx + C`

⇒ `y sec^2x = sec x + C`

When `x = pi/3, y = 0;  "then"  0 =  sec  pi/3 + C`

⇒ C = -2

Putting in (1), y sec² x = sec x - 2

⇒ y = cos x - 2 cos²x, 

Which is the required solution.