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Question
Solve the following:
`("d"y)/("d"x) + y/x = x"e"^x`
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Solution
It is of the form `("d"y)/("d"x) + "P"y` = Q
Here P = `1/x`
Q = xex
`int "Pd"x = int 1/x "d"x`
= log x
I.F = `"e"^(intpdx)`
= elog x
= x
The required solution is
y(I.F) = `int "Q"("I.F") "d"x + "c"`
y(x) = `int x"e"^x (x) "d"x`
xy = `int x^2"e"^x "d"x`
∴ xy = `x^2"e"^x - 2x"e"^x + 2"e"^x + "c"`
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