what are the order and degree respectively of the differential equation
`{(d^(4)y//dx^(4))^(3)}^(2//3)-7x(d^(3)y//dx^(3))^(2)=8` ?

Options

A. 3,2
B. 4,3
C. 4,2
D. 3,3

Explanation

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

\[ \left(\frac{d^4y}{dx^4}\right)^{\frac{3}{2}} - 7x\left(\frac{d^3y}{dx^3}\right)^2 = 8 \]

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

- The term \(\frac{d^4y}{dx^4}\) is the fourth derivative of \(y\) with respect to \(x\).
- The term \(\frac{d^3y}{dx^3}\) is the third derivative of \(y\) with respect to \(x\).

The highest derivative present is \(\frac{d^4y}{dx^4}\), which is of order 4.

### Step 2: Determine the order of the differential equation
The order of a differential equation is defined as the highest order of the derivative present in the equation.

- Since the highest derivative is \(\frac{d^4y}{dx^4}\), the order of the differential equation is **4**.

### 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 the given equation, the term \(\left(\frac{d^4y}{dx^4}\right)^{\frac{3}{2}}\) is not a polynomial because of the fractional exponent \(\frac{3}{2}\).
- However, if we consider the highest order derivative term alone, we can see that it is raised to the power of \(\frac{3}{2}\) and is not a polynomial term.

Since the degree is defined only for polynomial forms, we cannot assign a degree to this equation in its current form. Thus, we conclude that the degree is **not defined** or **not applicable**.

### Final Answer
- **Order**: 4
- **Degree**: Not defined