Solve by separation of variables dy -(y - 1)^2dx = 0.

Solve by separation of variables dy -(y – 1)^2dx = 0.

Given differential equation is:

(1) $\begin{equation*} dy - (y-1)^2 dx = 0 \end{equation*}$

We can separate the variables as follows:

(2) $\begin{equation*} \frac{dy}{(y-1)^2} = dx \end{equation*}$

Now, we can integrate both sides of the equation:

(3) $\begin{equation*} \int \frac{dy}{(y-1)^2} = \int dx \end{equation*}$

Integrating the left-hand side requires the substitution $u = y-1$ and $du/dy = 1$ , yielding:

(4) $\begin{equation*} \int \frac{dy}{(y-1)^2} = -\frac{1}{y-1} + C_1 \end{equation*}$

where $C_1$ is the constant of integration.

For the right-hand side, we can simply integrate with respect to $x$ to obtain:

(5) $\begin{equation*} \int dx = x + C_2 \end{equation*}$

where $C_2$ is another constant of integration.

Substituting back $u=y-1$ , we have:

(6) $\begin{equation*} \int \frac{dy}{(y-1)^2} = -\frac{1}{y-1} + C_1 = x + C_2 \end{equation*}$

Simplifying and solving for $y$ , we get:

(7) $\begin{equation*} y = \frac{1}{x + C_2} + 1 + C_1 \end{equation*}$

where $C_1$ and $C_2$ are constants of integration.

Therefore, the solution to the differential equation $dy - (y-1)^2dx = 0$ is:

(8) $\begin{equation*} y = \frac{1}{x + C} + 1 \end{equation*}$

where $C = C_1 - C_2$ is the constant of integration.