Write the order and degree of the following differential equations :
`((d^(2)y)/(dx^(2)))^(3)+((dy)/(dx))^(4)+7=0`.

Answer

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

\[ \left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^4 + 7 = 0 \]

we will follow these steps:

### Step 1: Identify the highest derivative
The first step is to identify the highest derivative present in the differential equation.

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

The highest derivative here is \(\frac{d^2y}{dx^2}\), which is a 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 differential equation is:

\[ \text{Order} = 2 \]

### Step 3: 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 equation, the highest order derivative \(\frac{d^2y}{dx^2}\) is raised to the power of 3. Therefore, the degree of the differential equation is:

\[ \text{Degree} = 3 \]

### Final Result
Thus, we conclude that:
- **Order**: 2
- **Degree**: 3