Solve each of the following initial value problems:

, y = 0 when

xi. , y = 0 when

Given and

This is a first order linear differential equation of the form

Here, P = 2 tan x and Q = sin x

The integrating factor (I.F) of this differential equation is,

We have

[∵ m log a = log a^m]

∴ I.F = sec²x [∵ e^{log x} = x]

Hence, the solution of the differential equation is,

Recall

⇒ ysec²x = sec x + c

∴ y = cos x + c cos²x

However, when, we have y = 0

∴ c = –2

By substituting the value of c in the equation for y, we get

y = cos x + (–2)cos²x

∴ y = cos x – 2cos²x

Thus, the solution of the given initial value problem is y = cos x – 2cos²x