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Question
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
Options
`x/"e"^x`
`"e"^x/x`
xex
ex
Solution
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is `"e"^x/x`.
Explanation:
The given differential equation is `("d"y)/("d"x) + y = (1 + y)/x`
⇒ `("d"y)/("d"x) = (1 + y)/x - y`
⇒ `("d"y)/("d"x) = 1/x + y((1 - x))/x`
⇒ `("d"y)/("d"x) - ((1 - x)/x)y = 1/x`
Here, P = `-((1 - x)/x)` and Q = `1/x`
∴ Integrating factor I.F = `"e"^(intPdx)`
= `"e"^(int (x - 1)/x "d"x)`
= `"e"^(int(1 - 1/x)"d"x)`
= `"e"^((x - logx))`
= `"e"^x . "e"^(-logx)`
= `"e"^x . "e"^(log 1/x)`
= `"e"^x . 1/x`
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