Question

Show that tan^–1x + tan^–1y = C is the general solution of the differential equation (1 + x²) dy + (1 + y²) dx = 0.

Answer 1

Given differential equation is:

(1 + x²) dy + (1 + y²) dx = 0

⇒ `dy/dx = (-(1 + y^2))/((1 + x^2))`

`1/((1 + y^2)) dy = (-1)/((1 + x^2)) dx`

Integrating both sides w.r.t. x

`int dy/(1 + y^2) = -int 1/(1 + x^2) dx`

⇒ tan^–1y = – tan^–1x + C

∴ tan^–1x + tan^–1y = C

Hence, tan^–1x + tan^–1y = C is the general solution of the given differential equation.

Hence Proved.