Question

The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.

Answer 1

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`