Find the particular solution of the following differential equation. given that when x = 2, y = π

Given: x = 2, y = π

To find: the particular solution of the differential equation

given:

It can be rewritten as,

Dividing throughout by x we get,

It is a homogenous differential equation,

Now let

Now differentiate this with respect to x, we get

Applying the product rule of differentiation, we get

Substituting these values in equation (i), we get

Now integrating on both sides, we get

⇒ log |cosec v-cot v|=-log |x|+C

Substituting the value of v, we get

But given x=2 when y=π, now substituting these values in above equation, we get

But , so above equation becomes

Substituting the corresponding values, we get

⇒log |1|+log |2|=C

But log 1=0, so above equation becomes,

C=log 2

Now substituting this value in equation (ii), we get

Cancelling the log on both sides, we get

Hence this is the particular solution of the differential equation