Write the order and degree of the following differential equations.
`[1+((dy)/(dx))^2]^(3/2)= k (d^2y)/(dx^2)`

Options

A. order=1, degree =1
B. order=1, degree =2
C. order=2, degree =1
D. order=2, degree =2

Answer

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

\[ [1 + \left(\frac{dy}{dx}\right)^2]^{\frac{3}{2}} = k \frac{d^2y}{dx^2} \]

we will follow these steps:

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

The highest order derivative present in the equation is \(\frac{d^2y}{dx^2}\), which is a second-order derivative.

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

\[ \text{Order} = 2 \]

### Step 3: Determine the degree of the differential equation
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 case, the highest order derivative is \(\frac{d^2y}{dx^2}\), which appears in the equation as:

\[ k \frac{d^2y}{dx^2} \]

This term is raised to the power of 1 (since it is not raised to any power). Therefore, the degree is:

\[ \text{Degree} = 1 \]

### Final Answer
Thus, for the given differential equation:

- **Order**: 2
- **Degree**: 1