The given differential equation is:

$\text{dy} - (y - 1)^2 \, \text{dx} = 0$

We can rewrite it as:

$\frac{\text{dy}}{\text{dx}} = (y - 1)^2$

This is a separable differential equation. Let's solve it step by step:

1. Rewrite and separate the variables:

$\frac{\text{dy}}{(y - 1)^2} = \text{dx}$

2. Integrate both sides:

The left-hand side involves integrating $\frac{1}{(y - 1)^2}$, and the right-hand side involves integrating $1$.

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

Left-hand side:

$\int \frac{1}{(y - 1)^2} \, \text{dy} = -\frac{1}{y - 1}$

Right-hand side:

$\int 1 \, \text{dx} = x + C$

where $C$ is the constant of integration.

3. Combine the results:

$-\frac{1}{y - 1} = x + C$

4. Solve for $y$ explicitly (if needed):

Multiply through by $-1$: $\frac{1}{y - 1} = -(x + C)$

Take the reciprocal: $y - 1 = \frac{1}{-(x + C)} = -\frac{1}{x + C}$

Add 1 to both sides: $y = 1 - \frac{1}{x + C}$

Final Solution:

$y = 1 - \frac{1}{x + C}$

where $C$ is an arbitrary constant.

Would you like me to explain any specific step in detail or explore related topics?

Related Questions:

1. What are separable differential equations, and how are they solved?
2. How can the constant of integration $C$ be determined if an initial condition is provided?
3. What are some real-life applications of this type of differential equation?
4. How can this solution be verified by substitution into the original equation?
5. Can we analyze the behavior of the solution as $x \to \infty$ or $x \to -C$?

Tip:

Always check if a differential equation is separable before trying other methods—it simplifies the solving process significantly!