The general solution of the differential equation $x\, dy + y\, dx + 2x^3 \, dx =0, $ is

$xy+\frac{1}{2}x^4=C$

The correct answer is option (2) : $xy+\frac{1}{2}x^4=C$

$x\, dy + y\, dx + 2x^3dx =0$

$⇒d(xy) +2x^3 dx=0⇒d(xy) +\frac{1}{2}d(x^4)=0$

On integration, we get

$xy +\frac{1}{2}x^4 = C, $ which is the required solution of the given differential equation.