Find the particul

Given,

(tan^-1 y – x)dy = (1 + y²)dx

⇒

Clearly this is a linear differential equation. Comparing with the standard form

Solution of such equation is given by:

x(I.F) = where I.F = integrating factor

We get P(y) = & Q(y) =

Integrating factor I.F is given :

∴ I.F =

We know that:

∴ I.F =

∴ Solution is given as:

⇒ …(1)

Where I =

Let tan^-1y = u

⇒ du =

∴ I =

Using integration by parts:

I =

∴ I =

⇒ I =

∴ solution is given using equation 1:

⇒ …(2)

As we have to find particular solution, so we need to find value of C.

Given: x = 0, y = 0

∴ putting the values in eq (2)

0 = tan^-1(0) – 1 +

⇒ C = 1

∴ particular solution is –