Question

Find the particular solution of the differential equation `dy/dx + 2y tan x = sin x` given that y = 0 when x =  `pi/3`

Answer 1

The given differential equation is,

`dy/dx + 2y tanx = sinx` .....(1)

The above is a linear differential equation of the form of `dy/dx + Py  = Q`

where P = 2 tan x; Q = sin x

Now, `If = e^(intPdx) = e^(int2tanxdx) = e^(2log sec x) = sec^2 x`

Now, the solution of (1) is given by

`y xx IF = int [Q xx IF]dx + C`

`=> ysec^2 x = int[sin x xx sec^2x] dx + C`

`=> y sec^2 x = intsecx.tan x dx + C`

when `x = pi/3 , y = 3`

0 = 2 + C

C = -2

Particular solution

`ysec^2x = sec x - 2`