`(dy)/(dx)+(3x^2)/(1+x^3)y=(sin^2x)/(1+x^3)`

Options

A. `y(1+x^(3)) = x + 1/2 sin 2x +C`
B. ` y (1+x^(3))= Cx + 1/2 sin 2x `
C. ` y(1+x^(3)) = Cx - 1/2 sin 2x`
D. ` y (1+x^(3)) = x/2 -1/4 sin 2x +C`

Answer

d

Explanation

Given , ` (dy)/(dx) +(3x^(2))/(1+x^(3))y = (sin^(2)x)/(1+x^(3))`
Since, it is a liner differetial equation
Here, `P = (3x^(2))/(1+x^(3))`
` :. IF = e^(int Pdx)=e^(log(1+x^(3)))= 1+x^(3)`
Thus, solution is
` y(1+x^(3)) = int (sin^(2)x)/(1+x^(3))(1+x^(3))dx`
` = int (1-cs 2x)/2 dx`
` :. y (1+^(3))=1/2 x - (sin2x)/4 +C`