If p and q are order and degree of differential equation `y^(2)((d^(2)y)/(dx^(2)))^(2)+3x((dy)/(dx))^(1/3) + x^(2)y^(2) = sin x, then`

Options

A. `p gt q`
B. `(p)/(q) = (1)/(2)`
C. `p = q`
D. `p lt q`

Explanation

To solve the given differential equation and determine the order \( p \) and degree \( q \), we will follow these steps:

### Step 1: Identify the given differential equation
The differential equation is:
\[ y^2 \left( \frac{d^2y}{dx^2} \right)^2 + 3x \left( \frac{dy}{dx} \right)^{\frac{1}{3}} + x^2 y^2 = \sin x \]

### Step 2: Rewrite the equation
To analyze the order and degree, we need to express the equation in a standard form. We can rearrange the equation as:
\[ y^2 \left( \frac{d^2y}{dx^2} \right)^2 + 3x \left( \frac{dy}{dx} \right)^{\frac{1}{3}} + x^2 y^2 - \sin x = 0 \]

### Step 3: Determine the order \( p \)
The order of a differential equation is determined by the highest derivative present. In this equation, the highest derivative is \( \frac{d^2y}{dx^2} \), which is a second derivative. Therefore, the order \( p \) is:
\[ p = 2 \]

### Step 4: Determine the degree \( q \)
The degree of a differential equation is determined by the power of the highest order derivative when the equation is expressed in polynomial form with respect to that derivative. In our equation, the term involving the highest derivative is \( \left( \frac{d^2y}{dx^2} \right)^2 \), which has a power of 2. Thus, the degree \( q \) is:
\[ q = 2 \]

### Step 5: Analyze the options
Now we can analyze the relationships between \( p \) and \( q \):
- We have \( p = 2 \) and \( q = 2 \).
- Therefore, \( p = q \).

### Conclusion
From the analysis, we conclude:
- \( p = 2 \)
- \( q = 2 \)
- The correct option is \( p = q \).