Solve: dy = cosx (2 – y cosecx)dx.

Given, A differential equation dy = cosx (2 – y cosecx)dx

To find: Find the solution of the given differential equation.

Explanation: We have dy = cosx (2 – y cosecx)dx.

We can write as,

So,

….(i)

Now, It is a form of the linear differential equation in the form,

When comparing the equation (i) with a linear differential equation, we get

P = cot x and Q = 2cosx

Since, The solution of the Linear differential equation is

I.F =

And,

I.F × y

So, The solution for the given linear differential equation is

I.F

I.F = e^{log sin x}

I.F = sin x

Now, The general solution is

y. sin x

Since, sin2x = 2sinx.cosx

y. sin x

Let 2x = t

On differentiating this, we get

2 dx = dt

y. sin x

y. sin x

substitute the value of t; we get

y. sin x

Hence, This is the solution of given Linear differential equation.