What is the order of the differential equation `(dx)/(dy) + int y dx = x^(3)` ?

Options

A. A. 1
B. B. 2
C. C. 3
D. D. Cannot be determined

Answer

To determine the order of the differential equation given by:

\[ \frac{dx}{dy} + \int y \, dx = x^3 \]

we will analyze the components of the equation step by step.

### Step 1: Identify the terms in the equation
The equation consists of:
- A derivative term: \(\frac{dx}{dy}\)
- An integral term: \(\int y \, dx\)
- A polynomial term: \(x^3\)

### Step 2: Understand the differentiation and integration
The term \(\frac{dx}{dy}\) indicates that we are differentiating \(x\) with respect to \(y\). On the other hand, \(\int y \, dx\) indicates that we are integrating \(y\) with respect to \(x\).

### Step 3: Analyze the relationship between variables
In this equation, \(y\) is treated as a function of \(x\), and \(x\) is treated as a function of \(y\). However, the integration and differentiation are not with respect to the same variable, which complicates the analysis.

### Step 4: Determine the order of the differential equation
The order of a differential equation is defined as the highest derivative present in the equation. In this case, we have:
- The first derivative \(\frac{dx}{dy}\) (which is a first-order derivative).
- The integral \(\int y \, dx\) does not contribute to the order since it does not involve a derivative of \(x\) with respect to \(y\).

Since the integration is with respect to \(x\) and differentiation is with respect to \(y\), we cannot determine a clear order for the equation. The presence of the integral complicates the relationship between \(x\) and \(y\).

### Conclusion
Thus, we conclude that we cannot determine the order of the differential equation due to the mixed nature of differentiation and integration.

### Final Answer
The order of the differential equation is **not determinable**.

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