The order and degree of the differential equation `((d^2y)/(dx^2))^3 + ("dy"/"dx")^4-xy=0` are respectively

Options

A. 2 and 4
B. 3 and 2
C. 4 and 5
D. 2 and 3

Answer

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

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

we will follow these steps:

### Step 1: Identify the highest derivative
The first step is to identify the highest derivative present in the equation. In this case, we have:

- \(\frac{d^2y}{dx^2}\) (the second derivative)
- \(\frac{dy}{dx}\) (the first derivative)

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

### 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 is \(\frac{d^2y}{dx^2}\), which is the second derivative, the order of the differential equation is:

\[ \text{Order} = 2 \]

### 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 after the equation has been made a polynomial in derivatives.

In our equation, the term involving the highest order derivative is \(\left(\frac{d^2y}{dx^2}\right)^3\). The power of this term is 3. Therefore, the degree of the differential equation is:

\[ \text{Degree} = 3 \]

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

- **Order**: 2
- **Degree**: 3