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Question
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
Solution
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is y = `1/4 x^2 + "c" . x^-2`.
Explanation:
The given differential equation is `x(dy)/("d"x) + 2y = x^2`
⇒ `("d"y)/("d"x) + 2/x y` = x.
Since, it is linear differential equation
∴ P = `2/x` and Q = x
Integrating factor I.F. = `"e"^(int Pdx)`
= `"e"^(int 2/x "d"x)`
= `"e"^(2logx)`
= `"e"^(log x^2)`
= x2
∴ Solution is `y xx "I"."F". = int "Q" xx "I"."F". "d"x + "c"`
⇒ `y . x^2 = int x . x^2 "d"x + "c"`
⇒ `y . x^2 = int x^3 "d"x + "c"`
⇒ `y . x^2 = 1/4 x^4 + "c"`
⇒ y = `1/4 x^2 + "c" . x^-2`
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