https://file.aiquickdraw.com/mathbot/file/1725508827061i2os6q9q.jpg

The differential equation provided is:

\[
2x^4 \frac{d^4 y}{dx^4} + 5x^2 \left( \frac{dy}{dx} \right)^3 - xy = 0
\]

To determine the **order** and **degree** of this differential equation:

### 1. **Order**:
- The order of a differential equation is the highest derivative present in the equation. 
- In this case, the highest derivative is \( \frac{d^4 y}{dx^4} \), which is the fourth derivative of \(y\).
- Therefore, the **order** of the differential equation is **4**.

### 2. **Degree**:
- The degree of a differential equation is the power of the highest order derivative, provided the equation is polynomial in the derivatives.
- In the given equation, the term \( \frac{d^4 y}{dx^4} \) appears linearly (its power is 1), and all other derivatives, such as \( \left( \frac{dy}{dx} \right)^3 \), are also raised to integer powers.
- Therefore, the **degree** of the differential equation is **1**.

### Conclusion:
- The order is **4** and the degree is **1**, which corresponds to option **(a) 4, 1**.

Would you like further details or clarifications?

### Related Questions:
1. What is the general form of a linear differential equation?
2. How do you solve a fourth-order linear differential equation?
3. What are the conditions for a differential equation to be considered exact?
4. Can the degree of a differential equation be non-integer?
5. What is the difference between partial and ordinary differential equations?

**Tip**: Always check if a differential equation is polynomial in its derivatives before determining the degree.

The order and degree of the differential equation 2x^4 d^4y/dx^4 + 5x^2 (dy/dx)^3 - xy = 0 are, respectively, (a) 4.1 (b) 3.1 (c) 1.4 (d) 1.3.

Learn how to determine the order and degree of the differential equation 2x^4 d^4y/dx^4 + 5x^2 (dy/dx)^3 - xy = 0. This problem involves analyzing the highest derivative and understanding the degree based on polynomial terms.

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