Question

tan^–1x + tan^–1y = c is the general solution of the differential equation ______.

Options

• `("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`

• `("d"y)/("d"x) = (1 + x^2)/(1 + y^2)`

• (1 + x²)dy + (1 + y²)dx = 0

• (1 + x²)dx + (1 + y²)dy = 0

Answer 1

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

Explanation:

Given equation is tan^–1x + tan^–1y = c

Differentiating w.r.t. x, we have

`1/(1 + x^2) + 1/(1 + y^2) * ("d"y)/("d"x)` = 0

⇒ `(1/(1 + y^2)) ("d"y)/("d"x) = -(1/(1 + x^2))`

⇒ `("d"y)/("d"x) = -((1 + y^2)/(1 + x^2))`

⇒ (1 + x²)dy = – (1 + y²)dx

⇒ (1 + x²)dy + (1 + y²)dx = 0.