`x((d^(3)y)/(dx^(3)))^(2)-((d^(2)y)/(dx^(2)))^(4)=0` . Find order and degree of differential equation.

Options

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

Answer

To find the order and degree of the given differential equation

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

we will follow these steps:

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

In the given equation, we have:
- \(\frac{d^3y}{dx^3}\) which is the third derivative of \(y\)
- \(\frac{d^2y}{dx^2}\) which is the second derivative of \(y\)

The highest order derivative is \(\frac{d^3y}{dx^3}\).

### 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 order derivative is \(\frac{d^3y}{dx^3}\), the order of the differential equation is:

**Order = 3**

### 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 expressed as a polynomial in derivatives.

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

**Degree = 2**

### Final Answer
Thus, the order of the differential equation is **3** and the degree is **2**.

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