An integrating factor of the differential equation $$(2x^2+y^2+x)dx+xy dy=0$$ is​

Correct option is D

Solution:

$$\text{Given differential equation:} \\\ \\(2x^2 + y^2 + x)\,dx + xy\,dy = 0 \\\ \\M(x, y) = 2x^2 + y^2 + x,\quad N(x, y) = xy \\\ \\\text{Check for exactness:} \\\ \\\frac{\partial M}{\partial y} = 2y,\quad \frac{\partial N}{\partial x} = y \\\ \\\Rightarrow \text{Not exact since } \frac{\partial M}{\partial y} \ne \frac{\partial N}{\partial x} \\\ \\\text{Try integrating factor } \mu(x) = x: \\\ \\\ \\x(2x^2 + y^2 + x)\,dx + x(xy)\,dy \\\ \\= (2x^3 + x y^2 + x^2)\,dx + x^2 y\,dy \\\ \\M = 2x^3 + x y^2 + x^2,\quad N = x^2 y \\\ \\\frac{\partial M}{\partial y} = 2x y,\quad \frac{\partial N}{\partial x} = 2x y \\\Rightarrow \text{Exact equation} \\\boxed{\text{Correct integrating factor is } \mu(x) = x}$$​