Find the order and degree (if defined) of the equation: `a=(1[1+((dy)/(dx))^2]^(3/2))/((d^2y)/(dx^2)),` where `a` is constant

Explanation

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

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

where \( a \) is a constant, we can follow these steps:

### Step 1: Rewrite the equation
First, we can rearrange the equation to isolate the second derivative:

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

### Step 2: Identify the highest derivative
In this equation, we have two derivatives:
- The first derivative \( \frac{dy}{dx} \)
- The second derivative \( \frac{d^2y}{dx^2} \)

The highest derivative present in the equation is \( \frac{d^2y}{dx^2} \).

### Step 3: 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 here is \( \frac{d^2y}{dx^2} \), the order of the equation is:

\[ \text{Order} = 2 \]

### Step 4: Determine the degree
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, we can express the equation in a polynomial form:

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

However, the right side involves a fractional power, which complicates the definition of degree. To find the degree, we need to express the equation in a polynomial form.

Squaring both sides gives:

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

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

\[ \text{Degree} = 2 \]

### Final Answer
Thus, the order and degree of the given differential equation are:

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