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The problem in the image asks to classify two ordinary differential equations (ODEs) as either homogeneous, first-order, and/or Bernoulli equations. Let's analyze both equations:

### (a) \(\frac{dy}{dx} + \frac{2x}{y^2} + \frac{y}{x} = 0\)

1. **First-order**: 
   - This is a first-order ODE since the highest derivative is \(\frac{dy}{dx}\).

2. **Homogeneous**:
   - To check if it's homogeneous, the degree of each term involving \(x\) and \(y\) should be the same. Let's rewrite the terms:
     - The term \(\frac{dy}{dx}\) has no direct degree involving \(x\) or \(y\).
     - The term \(\frac{2x}{y^2}\) has \(x^1 y^{-2}\), so the total degree is \(1 - 2 = -1\).
     - The term \(\frac{y}{x}\) has \(y^1 x^{-1}\), so the total degree is \(1 - 1 = 0\).
   - Since the terms do not have the same degree, this is **not homogeneous**.

3. **Bernoulli**:
   - A Bernoulli equation is of the form \(\frac{dy}{dx} + P(x)y = Q(x)y^n\). This equation doesn't fit the Bernoulli form because there is no power of \(y\) that matches this structure.
   
### Conclusion: Equation (a) is **first-order** but neither **homogeneous** nor **Bernoulli**.

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### (b) \((2y^2 + 4x^2) dx - xydy = 0\)

1. **First-order**:
   - This is also a first-order equation since there are no higher-order derivatives.

2. **Homogeneous**:
   - We check for homogeneity by expressing the equation in terms of degree:
     - The term \(2y^2 dx\) has \(y^2\), so its degree is 2.
     - The term \(4x^2 dx\) has \(x^2\), so its degree is 2.
     - The term \(-xy dy\) has \(x^1 y^1\), so its degree is \(1 + 1 = 2\).
   - Since all terms have degree 2, this equation is **homogeneous**.

3. **Bernoulli**:
   - This equation is not in the form of a Bernoulli equation, so it is **not Bernoulli**.

### Conclusion: Equation (b) is **first-order** and **homogeneous**, but not Bernoulli.

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Would you like further details or clarifications on these classifications? 

Here are 5 related questions to extend this topic:

1. How do you solve a first-order homogeneous differential equation?
2. What is the method to solve Bernoulli differential equations?
3. Can a differential equation be both homogeneous and Bernoulli?
4. How can you rewrite a non-homogeneous ODE into a homogeneous one?
5. What role does degree play in determining whether an equation is homogeneous?

**Tip**: For classification, always check the structure of the equation first before deciding on the method of solution.

Classify the following ODEs as homogeneous first order and/or Bernoulli.

Analyze and classify ordinary differential equations (ODEs) as first-order, homogeneous, or Bernoulli. Learn key mathematical concepts, including first-order ODEs and Bernoulli equations, with examples and classifications. Suitable for college-level math courses.

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