To solve this differential equation, we can first rearrange it to separate the variables x and y:

x^3y' + x^2y + x + 1 = 0
x^3y' + x^2y = -x - 1

Now, we can rewrite the equation in the form y' + $x/y$y = -x^2/y:

y' + $x/y$y = -x^2/y

This is a first-order linear differential equation. To solve it, we can use an integrating factor. The integrating factor is e^$\int (x/y)dx$, which simplifies to e^ln|y| = |y|.

Multiplying throughout the equation by the integrating factor, we get:

|y|y' + x|y| = -x^2

Let z = y^2, then z' = 2yy'. So, we have:

|y|z' + x|y| = -x^2

Now we can solve this new differential equation for z. We can rewrite the equation as:

z' = $x^2/|y|$ - x

Integrating both sides with respect to x, we get:

z = $\int (x^2/|y|)dx$ - ∫xdx
z = $\int (x^2/|y|)dx$ - $x^2/2$ + C

Thus, the general solution to the original differential equation is:

y^2 = $x^3/3$ln|x| - $x^2/2$x + C

where C is the constant of integration.