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Question
Solve: `("d"y)/("d"x) + 2/xy` = x2
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Solution
`("d"y)/("d"x) + 2/xy` = x2
The given equation is of the form
`("d"y)/("d"x) + "P"y` = Q.
where P = `2/x` and Q = x2
∴ I.F. = `"e"^(int^("Pd"x))`
= `"e"^(int2/x) "d"x`
= `"e"^(2logx)`
= `"e"^(log x^2)`
= x2
∴ Solution of the given equation is
`y("I"."F".) = int"Q"("I.""F.") "d"x + "c"_1`
∴ `y * x^2 = intx^2 * x^2 "d"x + "c"_1`
∴ `yx^2 = intx^4 "d"x + "c"_1`
∴ yx2 = `x^5/5 + "c"_1`
∴ 5x2y = x5 + c, where c = 5c1
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