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Question
Solve the following:
`x ("d"y)/("d"x) + 2y = x^4`
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Solution
The given equation can be reduced to
`("d"y)/("d"x) + (2y)/x = x^3`
It is of the form `("d"y)/("d"x) + "P"y` = Q
Here P = `2/x`, Q = x3
`int "pd"x = int 2/x "d"x`
= `2 int 1/x "d"x`
= `2 log x - log x^2`
I.F = `"e"^(int Pdx)`
= `"e"^(log x^2)`
= x2
The required solution is
y(I.F) = `int "Q" ("I.F") "d"x + "c"`
y(x2) = `int x^3 (x^2) "d"x + "c"`
x2y = `int x^5 "d"x + "c"`
x2y = `x^6/6 + "c"`
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