Question

The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx².

Options

• True

• False

Answer 1

This statement is True.

Explanation:

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

⇒ `("d"y)/("d"x) = 1 + 2 y/x`

⇒ `("d"y)/("d"x) = (2y)/x` = 1

Here, P = `(-2)/x` and Q = 1

Integrating factor I.F. = `"e"^(int(-2)/x "d"x)`

= `"e"^(-2 log x)`

= `"e"^(log x^-2)`

= `1/x^2`

∴ Solution is `y xx "I"."F". = int "Q" xx "I"."F". "d"x + "c"`

⇒ `y xx 1/x^2 = int 1 xx 1/x^2 "d"x + "c"`

⇒ `y/x^2 = int 1/x^2 "d"x + "c"`

⇒ `y/x^2 = - 1/x + "c"`

⇒ y = `-x + "c"x^2`

⇒ y + x = cx²