The order and degree of the differential equation.
`[ 1 + ((dy)/(dx))^(2)]^(3) = rho^(2) [ (d^(2) y)/(dx^(2))]^(2)` are respectively .

Options

A. 3 and 2
B. 2 and 2
C. 2 and 3
D. 1 and 3

Answer

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

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

we will follow these steps:

### Step 1: Identify the highest derivative present in the equation.
In the given equation, we can see that there are two derivatives:
- The first derivative \(\frac{dy}{dx}\)
- The second derivative \(\frac{d^{2}y}{dx^{2}}\)

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

### 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 derivative we identified is \(\frac{d^{2}y}{dx^{2}}\), the order of the differential equation is 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 equation, the term involving the highest order derivative \(\frac{d^{2}y}{dx^{2}}\) is raised to the power of 2. Thus, the degree of the differential equation is 2.

### Conclusion
Combining the results from Steps 2 and 3, we find that the order and degree of the differential equation are both 2.

Thus, the final answer is:
- Order: 2
- Degree: 2

### Final Answer:
Order and Degree: (2, 2)

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