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The differential equation provided is:

\[
dy - (y - 1)^2 dx = 0
\]

This can be rewritten as:

\[
dy = (y - 1)^2 dx
\]

To solve this using the method of separation of variables, follow these steps:

1. Separate the variables \( y \) and \( x \):
   \[
   \frac{dy}{(y - 1)^2} = dx
   \]

2. Integrate both sides. On the left side, we integrate with respect to \( y \), and on the right side, with respect to \( x \):

\[
\int \frac{dy}{(y - 1)^2} = \int dx
\]

The integral on the left side is:

\[
\int \frac{dy}{(y - 1)^2} = -\frac{1}{y - 1}
\]

The integral on the right side is:

\[
\int dx = x + C
\]

where \( C \) is the constant of integration.

3. Now, equate both integrals:

\[
-\frac{1}{y - 1} = x + C
\]

4. Solve for \( y \):

\[
\frac{1}{y - 1} = -(x + C)
\]

\[
y - 1 = \frac{1}{-(x + C)} = \frac{1}{C - x}
\]

\[
y = 1 + \frac{1}{C - x}
\]

Thus, the general solution to the differential equation is:

\[
y = 1 + \frac{1}{C - x}
\]

Let me know if you'd like any further details or have any questions.

Here are 5 related questions for practice:
1. How do you solve a differential equation using the integrating factor method?
2. What are the common types of first-order differential equations?
3. How do you interpret a separable differential equation graphically?
4. Can this differential equation be solved using another method like exact equations?
5. How can initial conditions be applied to find a specific solution to this equation?

**Tip:** Always check whether a differential equation is separable before trying more advanced methods, as separation of variables often simplifies the problem considerably.

Apply variable separation method to find the solution to: dy - (y - 1)^2 dx = 0

Learn how to solve the differential equation dy - (y - 1)^2 dx = 0 using the method of separation of variables. This solution includes a step-by-step approach to separating variables and integrating both sides. Suitable for college-level calculus students.

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