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Question
The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is ______.
Options
y =`"e"^(x - y) = x^2 "e"^-y + "c"`
`"e"^y - "e"^x = x^3/3 + "c"`
`"e"^x + "e"^y = x^3/3 + "c"`
`"e"^x - "e"^y = x^3/3 + "c"`
Solution
The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is `"e"^y - "e"^x = x^3/3 + "c"`.
Explanation:
The given differential equation is `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y`
⇒ `("d"y)/("d"x) = "e"^x . "e"^-y + x^2 . "e"^-y`
⇒ `("d"y)/("d"x) = "e"^-y ("e"^x + x^2)`
⇒ `("d"y)/"e"^-y = ("e"^x + x^2)"d"x`
⇒ `"e"^y . "d"y = ("e"^x + x^2)"d"x`
Integrating both sides, we have
`int "e"^x "d"y = int ("e"^x + x^2) "d"x`
⇒ `"e"^y = "e"^x + x^3/3 + "c"`
⇒ `"e"^y - "e"^x = x^3/3 + "c"`
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