Question

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

Options

• xe^x

• `(e^x)/x`

• `x/e^x`

• xe^1/x

Answer 1

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

Explanation:

Given, `(dy)/(dx) + y = (1 + y)/x`

`(dy)/(dx) + y = 1/x + 1/y`

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

Here, P = `1 - 1/x`

I.F. = `int_(e) pdx`

= `int_e(1 - 1/x)dx`

= `e^(x - logx)`

= `e^x.e^(log e 1/x)`

= `1/xe^x`