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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`
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Solution
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)`.
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