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Question
Solve the following:
`("d"y)/("d"x) - y/x = x`
Solution
Given `("d"y)/("d"x) + ((-1)/x)y = x`
It is of the form `("d"y)/("d"x) + "p"y` = Q
Here P = `(-1)/x, "Q"` = x
`int "p" "d"x = int (-1)/x "d"x`
= `- log x`
= `log(1/x)`
I.F = `"e"^(int pdx)`
= `"e"^(log(1/x)`
= `1/x`
The required solution is
y(I.F) = `int "Q"("I.F") "d"x + "c"`
`y(1/x) = int x(1/x) "d"x + "c"`
`y/x = int "d"x + "c"`
∴ `y/x` = x + c
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