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Question
Solve `x ("d"y)/(d"x) + 2y = x^4`
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Solution
`x ("d"y)/(d"x) + 2y = x^4`
÷ each term by x
`("d"y)/("d"x) + (2y)/x = x^2`
This is of the form `("d"y)/("d"x) + "P"y` = Q
Here P = `2/x` and Q = x3
`int "Pd"x = 2int1/x "d"x`
= 2 log x
= log x2
I.F = `"e"^(intpdx)`
=`"e"^(logx^2)`
= x2
This solution is
y(I.F) = `int "Q"x ("I.F") d"x + "c"`
y(x2) = `int (x^3 xx x^2) 'd"x + "c"`
yx2 = `int x^5 "d"x + "c"`
⇒ yx2 = `x^6/6 + "c"`
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