The order and degree of differential equation: `[1+((dy)/(dx))^(2)]=(d^(2)y)/(dx^(2)"are"`

Options

A. `2,(3)/(2)`
B. 2,3
C. 2,1
D. 3,4

Explanation

To determine the order and degree of the given differential equation:

\[ 1 + \left( \frac{dy}{dx} \right)^2 = \frac{d^2y}{dx^2} \]

we will follow these steps:

### Step 1: Identify the highest order derivative
The given differential equation contains two types of derivatives:
- The first derivative: \( \frac{dy}{dx} \)
- The second derivative: \( \frac{d^2y}{dx^2} \)

To find the order of the differential equation, we need to identify the highest order derivative present in the equation. Here, the highest order derivative is \( \frac{d^2y}{dx^2} \), which is the second derivative.

### Step 2: Determine the order
The order of a differential equation is defined as the highest order of derivative present in the equation. Since the highest derivative is the second derivative, the order of the given differential equation is:

**Order = 2**

### Step 3: Identify the degree of the highest order derivative
The degree of a differential equation is defined as the power of the highest order derivative when the equation is a polynomial in derivatives. In our equation, the highest order derivative \( \frac{d^2y}{dx^2} \) appears without any exponent other than 1.

### Step 4: Determine the degree
Since the highest order derivative \( \frac{d^2y}{dx^2} \) is raised to the power of 1, the degree of the differential equation is:

**Degree = 1**

### Final Result
Thus, the order and degree of the given differential equation are:
- **Order = 2**
- **Degree = 1**

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