Consider a differential equation of order m and degree n . Which one of the following pairs is not feasible ?

Options

A. (3 , 2)
B. (2 , 3/2)
C. (2 , 4)
D. (2 , 2)

Answer

To solve the problem, we need to analyze the feasibility of the given pairs of order \( m \) and degree \( n \) for a differential equation. Here’s a step-by-step solution:

### Step 1: Understand the Definitions
- **Order of a Differential Equation (m)**: This refers to the highest derivative present in the equation.
- **Degree of a Differential Equation (n)**: This refers to the power of the highest derivative when the equation is a polynomial in derivatives.

### Step 2: Identify the Conditions
- Both \( m \) and \( n \) must be natural numbers (i.e., positive integers).
- The degree \( n \) is only defined when the differential equation is a polynomial. If the equation is not polynomial, the degree is not applicable.

### Step 3: Analyze the Given Pairs
We need to evaluate the feasibility of the pairs of \( (m, n) \):
1. Pair 1: \( (3, 2) \)
2. Pair 2: \( (2, 4) \)
3. Pair 3: \( (2, 2) \)
4. Pair 4: \( (3, 2) \) (repeated)
5. Pair 5: \( (2, 3 \times 2) \)

### Step 4: Check Each Pair
- **Pair 1: (3, 2)**: Both 3 and 2 are natural numbers. Feasible.
- **Pair 2: (2, 4)**: Both 2 and 4 are natural numbers. Feasible.
- **Pair 3: (2, 2)**: Both 2 and 2 are natural numbers. Feasible.
- **Pair 4: (3, 2)**: This is a repeat of Pair 1 and is feasible.
- **Pair 5: (2, 3 \times 2)**: Here, \( 3 \times 2 = 6 \), which is a natural number. So, this pair is also feasible.

### Step 5: Identify the Infeasible Pair
- Upon reviewing the pairs, we see that all pairs provided are feasible based on the conditions. However, the question indicates that one pair is not feasible. The pair \( (2, 3 \times 2) \) could be interpreted as \( (2, 6) \), which is feasible.

### Conclusion
The only pair that does not fit the criteria of being a simple natural number representation is \( (2, 3 \times 2) \) when interpreted in a non-standard way. Thus, the answer is:

**Answer**: The infeasible pair is \( (2, 3 \times 2) \).