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Question
Solve the following:
`("d"y)/("d"x) + (3x^2)/(1 + x^3) y = (1 + x^2)/(1 + x^3)`
Solution
It is of the form `("d"y)/("d"x) + "P"y` = Q
Here P = `(3x^2)/(1 + x^3)`
Q = `(1 + x^2)/(1 + x^3)`
`int "Pd"x = int (3x^2)/(1 + x^3) "d"x`
= `log (1 + x^3)`
I.F = `"e"^(int Pdx)`
= `"e"^(log (1 + x^3))`
= `(1 + x^3)`
The required solution is
y(I.F) = `int "Q" ("I.F") "d"x + "c"`
`y(1 + x^3) = int ((1 + x^2))/((1 + x^3)) xx (1 + x^3) "d"x + "c"`
`y(1 + x^3) = int(1 + x^2) "d"x + "c"`
⇒ `y(1 + x^3) = x + x^3/3 + "c"`
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