What are the order and degree , respectively of the differential equation `((d^(3) y)/(dx^(3)))^(2) = y^(4) + ((dy)/(dx))^(5)` ?

Options

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

Answer

To determine the order and degree of the given differential equation \(\left(\frac{d^3 y}{dx^3}\right)^2 = y^4 + \left(\frac{dy}{dx}\right)^5\), 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 equation, we have:
- The term \(\frac{d^3 y}{dx^3}\) which is the third derivative of \(y\).

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

Since the highest derivative here is \(\frac{d^3 y}{dx^3}\), which is of order 3, we conclude that:
- **Order = 3**

### Step 3: Identify 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 term \(\left(\frac{d^3 y}{dx^3}\right)^2\) is raised to the power of 2.
- The other terms \(y^4\) and \(\left(\frac{dy}{dx}\right)^5\) do not affect the degree calculation since they do not involve the highest order derivative.

Thus, the degree of the differential equation is:
- **Degree = 2**

### Final Answer
Therefore, the order and degree of the given differential equation are:
- **Order = 3**
- **Degree = 2**

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