This is a first-order differential equation, specifically a separable differential equation, as it can be written in terms of \( y \) and \( x \) independently.

Given: 

\[
\frac{dy}{dx} = (y - 1)^2
\]
with the initial condition \( y(0) = 1 \).

### Step 1: Separate variables
We rewrite the equation to separate \( y \) and \( x \):

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

### Step 2: Integrate both sides
Now, we integrate both sides. First, integrate the left-hand side with respect to \( y \) and the right-hand side with respect to \( x \):

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

The integral on the left-hand side is:

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

On the right-hand side, we get:

\[
\int dx = x + C
\]

So, after integrating, we have:

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

### Step 3: Solve for \( y \)
Now, solve for \( y \). First, multiply both sides by \(-1\):

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

Take the reciprocal of both sides:

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

Thus:

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

### Step 4: Apply the initial condition \( y(0) = 1 \)
Substitute \( x = 0 \) and \( y = 1 \) into the equation to find \( C \):

\[
1 = 1 - \frac{1}{0 + C}
\]

This simplifies to:

\[
\frac{1}{C} = 0
\]

This is a problem because it implies that \( C \to \infty \), which suggests that the solution is constant \( y = 1 \) for all \( x \).

### Final Solution:
Thus, the solution to the differential equation is \( y(x) = 1 \), which satisfies both the differential equation and the initial condition.

Do you want more details or have any questions?

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Here are five related questions:
1. How would the solution change if the initial condition were \( y(0) = 2 \)?
2. What is a separable differential equation?
3. How do initial conditions affect the solutions of differential equations?
4. Can a differential equation have multiple solutions?
5. How can differential equations model real-world phenomena?

**Tip:** Always verify initial conditions when solving differential equations—they determine the specific solution among possible general solutions.

Explore how to solve the separable differential equation dy/dx = (y - 1)^2 with the initial condition y(0) = 1. This step-by-step solution involves integrating both sides and applying initial conditions to arrive at the final solution, y(x) = 1. Suitable for undergraduate students studying differential equations.

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