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Let's analyze the equation \( y' = \frac{xy - 3y^2}{x^2} \) to determine whether it is homogeneous.

### Step 1: Determine Homogeneity
An equation is homogeneous if it can be written in the form:
\[
y' = \frac{f(x, y)}{g(x, y)}
\]
where both \( f(x, y) \) and \( g(x, y) \) are homogeneous functions of the same degree.

The given differential equation is:
\[
y' = \frac{xy - 3y^2}{x^2}
\]

We will check if the numerator \( xy - 3y^2 \) and the denominator \( x^2 \) are homogeneous functions of the same degree.

- The term \( xy \) is of degree 2, because \( \text{degree}(x) = 1 \) and \( \text{degree}(y) = 1 \), so \( \text{degree}(xy) = 1 + 1 = 2 \).
- The term \( -3y^2 \) is of degree 2, since \( \text{degree}(y^2) = 2 \).
- The denominator \( x^2 \) is also of degree 2.

Since all terms (both in the numerator and denominator) are homogeneous of degree 2, the differential equation is **homogeneous**.

### Step 2: Solve the Homogeneous Differential Equation
For homogeneous differential equations, we perform the substitution:
\[
y = vx \quad \text{and} \quad y' = v + x \frac{dv}{dx}
\]
Substitute \( y = vx \) into the given equation:
\[
v + x \frac{dv}{dx} = \frac{x(vx) - 3(vx)^2}{x^2}
\]
Simplify the right-hand side:
\[
v + x \frac{dv}{dx} = \frac{x^2v - 3x^2v^2}{x^2}
\]
\[
v + x \frac{dv}{dx} = v - 3v^2
\]

Cancel the \( v \) terms:
\[
x \frac{dv}{dx} = -3v^2
\]

### Step 3: Separate Variables
Now we separate the variables \( v \) and \( x \):
\[
\frac{dv}{v^2} = -3 \frac{dx}{x}
\]

Integrating both sides:
\[
\int v^{-2} \, dv = -3 \int \frac{1}{x} \, dx
\]
\[
\frac{1}{v} = -3 \ln|x| + C
\]

### Step 4: Back-substitute \( v = \frac{y}{x} \)
\[
\frac{x}{y} = -3 \ln|x| + C
\]

Finally, solve for \( y \):
\[
y = \frac{x}{-3 \ln|x| + C}
\]

Thus, the general solution to the homogeneous differential equation is:
\[
y = \frac{x}{-3 \ln|x| + C}
\]

Let me know if you have questions or want additional clarification. Here are some related questions you might consider:

1. What are the conditions for a differential equation to be classified as homogeneous?
2. How does the substitution \( y = vx \) simplify the process of solving homogeneous differential equations?
3. Can all homogeneous differential equations be solved using the method of separation of variables?
4. How do boundary conditions affect the solution of a homogeneous differential equation?
5. What are the advantages of using logarithmic expressions in the solutions of differential equations?

**Tip:** Always check the degrees of all terms in both the numerator and denominator when determining if a differential equation is homogeneous.

Verifica si la ecuación diferencial y' = (xy - 3y^2) / x^2 es o no homogénea. En caso de ser homogénea, resolverla.

Learn how to determine if the differential equation y' = (xy - 3y^2) / x^2 is homogeneous and solve it using substitution and separation of variables. This step-by-step guide covers key concepts in differential equations for undergraduate mathematics students.

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