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Solve the given differential equation by separation of variables.

The given differential equation is:

\[
\frac{dy}{dx} = \frac{xy + 5x - y - 5}{xy - 2x + 6y - 12}
\]

This equation does not appear to be directly separable in its current form. In order to check whether separation of variables can be applied, we need to simplify and manipulate the equation. Let’s explore step-by-step whether we can rewrite the expression in a separable form.

1. First, we can try to factor the numerator and denominator to see if common terms can be canceled or if the equation simplifies.

**Numerator**: \( xy + 5x - y - 5 \)

We can factor by grouping:
\[
xy + 5x - y - 5 = x(y + 5) - 1(y + 5) = (x - 1)(y + 5)
\]

**Denominator**: \( xy - 2x + 6y - 12 \)

Similarly, factor by grouping:
\[
xy - 2x + 6y - 12 = x(y - 2) + 6(y - 2) = (x + 6)(y - 2)
\]

Now the equation becomes:
\[
\frac{dy}{dx} = \frac{(x - 1)(y + 5)}{(x + 6)(y - 2)}
\]

2. We can now attempt to separate the variables. Rearranging the equation to separate \(x\) and \(y\):
\[
\frac{y + 5}{y - 2} \, dy = \frac{x - 1}{x + 6} \, dx
\]

3. Integrate both sides:

On the left-hand side, we integrate with respect to \(y\):
\[
\int \frac{y + 5}{y - 2} \, dy
\]

This can be split into:
\[
\int \left( 1 + \frac{7}{y - 2} \right) dy = \int 1 \, dy + \int \frac{7}{y - 2} \, dy = y + 7 \ln |y - 2|
\]

On the right-hand side, integrate with respect to \(x\):
\[
\int \frac{x - 1}{x + 6} \, dx
\]

Similarly, this can be split:
\[
\int \left( 1 - \frac{7}{x + 6} \right) dx = \int 1 \, dx - \int \frac{7}{x + 6} \, dx = x - 7 \ln |x + 6|
\]

4. Combine the results from both sides:
\[
y + 7 \ln |y - 2| = x - 7 \ln |x + 6| + C
\]

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

Thus, the solution to the differential equation is:
\[
y + 7 \ln |y - 2| = x - 7 \ln |x + 6| + C
\]

This is the implicit solution. Further steps would require solving for \(y\) explicitly, if necessary, which may not be easily achievable in this case.

Do you want more details on any specific part of the process, or help with further manipulation?

Here are 5 related questions to explore:
1. Can we express \(y\) explicitly from the solution?
2. How do we handle constants of integration when solving differential equations?
3. What conditions would make a differential equation easily separable?
4. Could we apply another method, such as an integrating factor, to solve this equation?
5. What is the role of logarithmic terms in the solution process?

**Tip**: When solving differential equations, always check if factoring or algebraic simplifications can make the equation separable!

Solve the given differential equation by separation of variables: dy/dx = (xy + 5x - y - 5)/(xy - 2x + 6y - 12)

Learn how to solve a differential equation through separation of variables by factoring the equation into simpler terms. The solution involves integrating both sides, leading to an implicit solution. This step-by-step process is ideal for students at an advanced high school or undergraduate level.

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