The order and degree of the differential equation `((d^(2)y)/(dx^(2)))^(1/6) - ((dy)/(dx))^(1/3) =0` are respectively .

Options

A. 3,2
B. 2,3
C. 6,3
D. 3,1

Answer

To determine the order and degree of the differential equation

\[ \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{6}} - \left(\frac{dy}{dx}\right)^{\frac{1}{3}} = 0, \]

we will follow these steps:

### Step 1: Identify the highest order derivative
The given differential equation contains two derivatives:

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

The highest order derivative present is \(\frac{d^2y}{dx^2}\), which is a second derivative.

### 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 order derivative is the second derivative, the order of the differential equation is:

\[ \text{Order} = 2. \]

### Step 3: Simplify the equation to find the degree
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.

To express the equation in a polynomial form, we can manipulate it as follows:

1. Start with the original equation:

\[ \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{6}} = \left(\frac{dy}{dx}\right)^{\frac{1}{3}}. \]

2. Raise both sides to the power of 6 to eliminate the fractional exponents:

\[ \left(\frac{d^2y}{dx^2}\right) = \left(\frac{dy}{dx}\right)^2. \]

### Step 4: Identify the degree
Now we can see that the highest order derivative \(\frac{d^2y}{dx^2}\) appears to the power of 1. Therefore, the degree of the differential equation is:

\[ \text{Degree} = 1. \]

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

- **Order:** 2
- **Degree:** 1