To solve the given equation x^3 dx + (y - 1)^2 dy = 0, we can use the concept of exact equations. An exact equation is a first-order differential equation of the form M(x, y) dx + N(x, y) dy = 0, where the total differential of a function u(x, y) satisfies du = M dx + N dy.

In order to check if the given equation is exact, we need to verify if the partial derivatives of M and N with respect to y and x, respectively, are equal. Let's calculate these partial derivatives:

∂M/∂y = 0
∂N/∂x = 3x^2

Since ∂M/∂y is not equal to ∂N/∂x, the given equation is not exact. However, we can make it exact by multiplying through by an integrating factor.

Step 1: Find the integrating factor
We can find the integrating factor (IF) by dividing the partial derivative of N with respect to x by the partial derivative of M with respect to y and then integrating with respect to x. The formula for the integrating factor is given by:

IF = e^(∫(∂N/∂x - ∂M/∂y) / M) dx

In this case, we have:
IF = e^(∫(3x^2 - 0) / x^3) dx
= e^(∫3/x dx)
= e^(3ln|x| + C)
= e^(ln(x^3) + C)
= x^3e^C

Step 2: Multiply through by the integrating factor
Multiplying the given equation by the integrating factor, we get:
x^3(x^3e^C) dx + x^3(y - 1)^2 dy = 0

Simplifying this equation, we have:
x^6 e^C dx + x^3(y - 1)^2 dy = 0

Step 3: Integrate to find the solution
Integrating both sides of the equation, we get:
∫(x^6 e^C) dx + ∫(x^3(y - 1)^2) dy = ∫0 dx + ∫0 dy

Simplifying the integrals, we have:
(x^7 e^C)/7 + (x^4(y - 1)^3)/4 = c

This is the general solution to the given equation. However, it does not match any of the provided answer options. It seems there may be an error in the given options or the question itself.