To solve the differential equation \(\frac{dx}{dy} = \left(\frac{y}{x}\right)^2 + \frac{1}{\frac{y}{x} - 1}\), we need to simplify and then integrate.

First, let's rewrite the given equation in a more manageable form:

\[
\frac{dx}{dy} = \left(\frac{y}{x}\right)^2 + \frac{1}{\frac{y}{x} - 1}
\]

Simplify the terms:

\[
\frac{dx}{dy} = \frac{y^2}{x^2} + \frac{1}{\frac{y}{x} - 1}
\]

For the second term, \(\frac{1}{\frac{y}{x} - 1}\):

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

Now, the differential equation is:

\[
\frac{dx}{dy} = \frac{y^2}{x^2} + \frac{x}{y - x}
\]

Next, we separate the variables \(x\) and \(y\). However, this form doesn't seem to easily separate. Alternatively, we can check for any substitutions or a better approach. Let's consider \(u = \frac{y}{x}\), which simplifies our differential equation:

If \(u = \frac{y}{x}\), then \(y = ux\), and differentiating both sides with respect to \(y\):

\[
\frac{dy}{dx} = u + x \frac{du}{dx}
\]

Since we are given \(\frac{dx}{dy}\), let's consider the inverse \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} = \frac{1}{\left(\frac{y}{x}\right)^2 + \frac{1}{\frac{y}{x} - 1}}
\]

Using our substitution \(u = \frac{y}{x}\):

\[
\frac{dy}{dx} = \frac{1}{u^2 + \frac{1}{u - 1}}
\]

This is still quite complex, indicating the need for a numerical solution or further simplification may be required for exact integration.

If further steps or clarification is needed, please let me know!

Would you like a detailed step-by-step solution for this problem or do you have any specific questions?

Here are some topics you might be interested in asking about:
1. How to perform a substitution in differential equations?
2. Methods to solve first-order differential equations.
3. Examples of separable differential equations.
4. How to use integrating factors in solving differential equations?
5. The role of initial conditions in solving differential equations.

**Tip:** Always double-check the given differential equation for possible simplifications or substitutions that can make the problem more manageable.

Learn how to solve the differential equation (dx/dy) = (y/x)^2 + 1 / y/x - 1 with a detailed step-by-step solution. This problem involves the substitution method in differential equations and requires advanced understanding of calculus concepts. Suitable for advanced undergraduate students.

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