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Question
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
Solution
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is `"e"^x . 1/x`.
Explanation:
The given differential equation is `("d"y)/("d"x) + y = (1 + y)/x`
⇒ `("d"y)/("d"x) + y = (1 + y)/x`
⇒ `("d"y)/("d"x) + y = 1/x + y/x`
⇒ `("d"y)/("d"x) + y - y/x = 1/x`
⇒ `("d"y)/("d"x) + (1 - 1/x) = 1/x`
Here P = `(1 - 1/x)`
∴ I.F. = `"e"^(intPdx)`
= `"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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