Question

The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.

Options

• `x/"e"^x`

• `"e"^x/x`

• xe^x 

• e^x 

Answer 1

The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is `"e"^x/x`.

Explanation:

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

⇒ `("d"y)/("d"x) = (1 + y)/x - y`

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

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

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

∴ Integrating factor I.F = `"e"^(intPdx)`

= `"e"^(int (x - 1)/x "d"x)`

= `"e"^(int(1 - 1/x)"d"x)`

= `"e"^((x - logx))`

= `"e"^x . "e"^(-logx)`

= `"e"^x . "e"^(log  1/x)`

= `"e"^x . 1/x`