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### Part 1: Classification of Differential Equations
We need to determine the classification of each differential equation based on **type, order, degree**, and **linearity**.

1. **Equation 1:**
   \[
   y' = \sqrt{y'' + \sqrt{y'' + \sqrt{y'' + \sqrt{y'' + \ldots}}}}
   \]
   - **Order**: The highest derivative is \( y'' \) (second derivative), so this is a **second-order** differential equation.
   - **Degree**: The equation involves square roots, which makes the degree **non-polynomial** (since roots of derivatives are non-integer powers).
   - **Linearity**: The equation is **non-linear** due to the presence of square roots and combinations of derivatives.
   
2. **Equation 2:**
   \[
   (y^{(4)} - 1)(y^{(4)})^2 + y^{(4)} + 1) (y^{(4)} + 5) = 0
   \]
   - **Order**: The highest derivative is \( y^{(4)} \) (fourth derivative), so this is a **fourth-order** differential equation.
   - **Degree**: The degree is **2**, since the highest power of the highest derivative \( y^{(4)} \) is squared.
   - **Linearity**: The equation is **non-linear** because of the product of the terms involving the fourth derivative \( y^{(4)} \).

3. **Equation 3:**
   \[
   xy'' + x^2y' + \cos(x)y = \ln(x + \cos(x))
   \]
   - **Order**: The highest derivative is \( y'' \) (second derivative), so this is a **second-order** differential equation.
   - **Degree**: The degree is **1** (as all terms are linear with respect to the highest derivative \( y'' \)).
   - **Linearity**: The equation is **linear** because there are no products or non-linear functions of the derivatives \( y', y'' \).

### Part 2: Verify the Solution of the Differential Equation

4. **Equation 4:**
   \[
   y'' - y' - y = e^x + 5x^2 e^{-2x} - 11xe^{-2x} + 2e^{-2x}
   \]
   With the given solution \( y = xe^x + x^2 e^{-2x} \), we need to substitute this into the differential equation and check if it satisfies the equation.

5. **Equation 5:**
   \[
   y'' + 3y' - 2y = \frac{x^2 + 4x + 3}{x^2 - 1}
   \]
   With the given solution \( y = \frac{x^2 + 4x + 3}{x^2 - 1} \), we will need to verify if it satisfies the equation by substitution.

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Let me solve Part 2 (Equations 4 and 5) to determine if the given solutions satisfy the differential equations. I'll proceed with that next.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can still provide a conceptual explanation of the problems or help with any other questions you might have! 

Would you like a breakdown of the method to manually verify these equations? Let me know if you'd like further clarification or have any questions. 

Here are some relative questions that might interest you:
1. How do we calculate the degree of non-linear differential equations?
2. What makes a differential equation linear or non-linear?
3. Can all higher-order differential equations be converted into a system of first-order equations?
4. How can you tell when a solution satisfies a given differential equation?
5. What are some real-world applications of solving non-linear differential equations?

**Tip:** When verifying solutions to differential equations, always calculate derivatives carefully and substitute them back into the original equation for a consistency check.

Determine the classification of the differential equation based on its types, order, degree, and linearity.

Learn how to classify differential equations based on their order, degree, and linearity with examples including equations with second and fourth derivatives. Suitable for advanced math students.

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