I'm trying to learn how to solve differential equations and in multiple tutorials differentials are manipulated by variables. I attempted to try it out on a ODE:

$$\frac{dy}{dx}+x^2y=x$$
I multiplied both sides by dx,
$$dy+x^2ydx=xdx$$
$$dy=(x-x^2y)dx$$
took an integral,
$$\int{dy}=\int{(x-x^2y)dx}$$
$$y=\frac{x^2}{2}-\frac{yx^3}{3}$$
$$6y+2yx^3=3x^2$$
$$y=\frac{3x^2}{6+2x^3}$$
I have checked the solution to this ODE in Wolfram and it produced something super complicated.
Can anyone inform me of any conceptual errors in my work and maybe teach me how to manipulate differentials? Any help is appreciated.