Question

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

Options

• y = `"ce"^((-x^2)/2`

• y = `"ce"^((x^2)/2`

• y = `(x + "c")"e"^((x^2)/2`

• y = `("c" - x)"e"^((x^2)/2`

Answer 1

The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is y = `(x + "c")"e"^((x^2)/2`.

Explanation:

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

⇒ `("d"y)/("d"x) - xy = "e"^((x^2)/2`

Since it is linear differential equation

Where P = –x and Q = `"e"^((x^2)/2`

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

= `"e"^(int -x  "d"x)`

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

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

⇒ `y xx "e"^( x^2/2) = int "e"^(x^2/2) "e"^(- x^2/2)  "d"x + "c"`

⇒ `y xx "e"^(- x^2/2) = int "e"^0  "d"x + "c"`

⇒ `y xx "e"^(- x^2/2) = int 1 . "d"x + "c"`

⇒ `y xx "e"^(- x^2/2) = x + "c"`

∴ y = `(x + "c")"e"^(x^2/2)`.