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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
Solution
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"`
⇒ ey = et + c
⇒ ey = `"e"^(y^2) + "c"`
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