Question

The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.

Options

• y = tan^–1x

• y – x = k(1 + xy)

• x = tan^–1y

• tan(xy) = k

Answer 1

The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is y – x = k(1 + xy).

Explanation:

The given differential equation is `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)`

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

Integrating both sides, we get

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

⇒ tan^–1y = tan^–1x + c

⇒ tan^–1y – tan^–1x = c

⇒ `tan^-1((y - x)/(1 + xy))` = c

⇒ `(y - x)/(1 + xy)` = tan c

⇒ `((y - x)/(1 + xy))` = k  ....[k = tan c]

⇒ y – x = k(1 + xy)