Question

The solution of `x ("d"y)/("d"x) + y` = e^x is ______.

Options

• y = `"e"^x/x + "k"/x`

• y = xe^x + cx

• y = xe^x + k

• x = `"e"^y/y + "k"/y`

Answer 1

The solution of `x ("d"y)/("d"x) + y` = e^x is y = `"e"^x/x + "k"/x`.

Explanation:

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

⇒ `("d"y)/("d"x) + y/x = "e"^x/x`

Here P = `1/x` and Q = `"e"^x/x`

∴ Integrating factor I.F. = `"e"^(int 1/x "d"x)`

= `"e"^(log |x|)`

= x

So, the solution is `y xx "I"."F". = int "Q" xx "I"."F". "d"x + "k"`

⇒ `y xx x = int "e"^x/x xx x  "d"x + "k"`

⇒ `y xx x = int "e"^x  "d"x + "k"`

⇒ `y xx x = "e"^x + "k"`

∴ y = `"e"^x/x + "k"/x`