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