Given the differential equation:

\[
\frac{dy}{dx} = \frac{2y^3}{x^2 y^2 - xy^2}
\]

Let's simplify the right-hand side of the equation before solving it.

### Simplification:

The right-hand side can be rewritten as:

\[
\frac{dy}{dx} = \frac{2y^3}{y^2 \cdot (x^2 - x)}
\]

This simplifies to:

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

So, the differential equation becomes:

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

This is a separable differential equation, meaning we can separate the variables \( y \) and \( x \) on different sides of the equation.

### Separation of Variables:

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

Now, integrate both sides:

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

### Left Side Integration:

\[
\int \frac{1}{y} \, dy = \ln |y|
\]

### Right Side Integration:

The right-hand side requires partial fraction decomposition:

\[
\frac{2}{x(x-1)} = \frac{A}{x} + \frac{B}{x-1}
\]

Solving for \( A \) and \( B \):

\[
2 = A(x-1) + Bx
\]

Substituting \( x = 1 \):

\[
2 = B(1) \implies B = 2
\]

Substituting \( x = 0 \):

\[
2 = A(-1) \implies A = -2
\]

So,

\[
\frac{2}{x(x-1)} = \frac{-2}{x} + \frac{2}{x-1}
\]

Thus:

\[
\int \frac{2}{x(x-1)} \, dx = \int \left( \frac{-2}{x} + \frac{2}{x-1} \right) dx = -2 \ln|x| + 2 \ln|x-1|
\]

### Putting it all together:

\[
\ln |y| = -2 \ln |x| + 2 \ln |x-1| + C
\]

Where \( C \) is the constant of integration. This can be further simplified as:

\[
\ln |y| = \ln \left| \frac{(x-1)^2}{x^2} \right| + C
\]

Exponentiate both sides:

\[
|y| = e^C \cdot \frac{(x-1)^2}{x^2}
\]

Let \( e^C = C_1 \), where \( C_1 \) is a new constant:

\[
y = \frac{C_1 (x-1)^2}{x^2}
\]

This is the general solution to the differential equation.

Would you like further details on any step or have any questions?

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Here are some related questions:

1. How do we solve differential equations that are not separable?
2. What are the different methods of solving first-order differential equations?
3. How do you apply boundary conditions to find a specific solution?
4. What is the significance of the constant of integration in the general solution?
5. How can partial fractions be used in different types of integrals?

**Tip:** When dealing with differential equations, always check if the equation is separable or if another method like an integrating factor might be needed.

Learn how to solve the separable differential equation dy/dx = 2y^3 / (x^2 y^2 - xy^2) step-by-step with detailed explanations. This problem covers differential equations, separable forms, and partial fraction decomposition methods. Suitable for advanced undergraduate students and those interested in differential equations.

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