Question

The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.

Options

• `"e"^(x^2 - y)` = c

• `"e"^-y + "e"^(x^2)` = c

• `"e"^-y = "e"^(x^2)` + c

• `"e"^(x^2 + y)` = c

Answer 1

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

Explanation:

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

⇒ `("d"y)/("d"x) = 2x . "e"^(x^2) . "e"^-y`

⇒ `("d"y)/("e"^-y) = 2x . "e"^(x^2)  "d"x`

Integrating both sides, we have

`int ("d"y)/("e"^-y) = int 2x . "e"^(x^2)  "d"x`

⇒ `int "e"^y  "d"y = int 2x . "e"^(x^2)  "d"x`

Pit in R.H.S. x² = t

∴ 2x dx = dt

∴ `int "e"^y  "d"y = int "e"^"t"  "dt"`

⇒ e^y = e^t + c

⇒ e^y = `"e"^(y^2) + "c"`