Consider the differential equation `[1 + ((dy)/(dx))^(2)]^(-3//2) = k ((d^2 y)/(dx^2))]`
The degree and order of the equation is :

Options

A. (2,2)
B. (3,2)
C. (2,3)
D. None of these

Answer

To find the degree and order of the given differential equation

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

we will follow these steps:

### Step 1: Identify the Order of the Differential Equation
The order of a differential equation is defined as the highest derivative present in the equation.

In the given equation, we see:
- The first derivative \(\frac{dy}{dx}\)
- The second derivative \(\frac{d^2y}{dx^2}\)

The highest derivative here is \(\frac{d^2y}{dx^2}\), which is of order 2.

**Order = 2**

### Step 2: Identify the Degree of the Differential Equation
The degree of a differential equation is defined as the power of the highest derivative after the equation has been expressed in a polynomial form in terms of the derivatives.

To analyze the degree, we first rewrite the equation in a more manageable form. We can square both sides to eliminate the fractional exponent:

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

Now, let's rewrite it:

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

Now, we can see that the highest derivative \(\frac{d^2y}{dx^2}\) is raised to the power of 2 in the equation.

Thus, the degree of the differential equation is 2.

**Degree = 2**

### Final Answer
- **Order:** 2
- **Degree:** 2