Question

The solution of `(1 + x^2) ("d"y)/("d"x) + 2xy - 4x^2` = 0 is ______.

Answer 1

The solution of `(1 + x^2) ("d"y)/("d"x) + 2xy - 4x^2` = 0 is y = `4/3 x^3/((1 + x^2)) + "c" (1 + x^2)^-1`.

Explanation:

The given differential equation is `(1 + x^2) ("d"y)/("d"x) + 2xy - 4x^2` = 0

⇒ `("d"y)/("d"x) + (2xy)/(1 + x^2) = (4x^2)/(1 + x^2)`

Since it is a linear differential equation

∴ P = `(2x)/(1 + x^2)` and Q = `(4x^2)/(1 + x^2)`

Integrating factor I.F. = `"e"^(int Pdx)`

= `"e"^(int (2x)/(1 + x^2) "d"x)`

= `"e"^(log(1 + x^2))`

= `(1 + x^2)`

∴ Solution is `y xx "I"."F" = int "Q" xx "I"."F". "d"x + "c"`

⇒ `y xx (1 + x^2) = int (4x)/(1 + x^2) xx (1 + x^2)"d"x + "c"`

⇒ `y xx (1 + x^2) = int 4x^2 "d"x + "c"`

⇒ `y xx (1 + x^2) = 4/3 x^3 + "c"`

⇒ y = `4/3 x^3/((1 + x^2)) + "c"(1 + x^2)^-1`