What are the degree and order respectively for the differential equation y = `x ((dy)/(dx))^(2) + ((dx)/(dy))^(2) `?

Options

A. 1 , 2
B. 2 , 1
C. 1 , 4
D. 4 , 1

Answer

To determine the degree and order of the given differential equation \( y = x \left( \frac{dy}{dx} \right)^2 + \left( \frac{dx}{dy} \right)^2 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ y = x \left( \frac{dy}{dx} \right)^2 + \left( \frac{dx}{dy} \right)^2 \]
We need to express \( \frac{dx}{dy} \) in terms of \( \frac{dy}{dx} \). Recall that:
\[ \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} \]
Thus, we can rewrite \( \left( \frac{dx}{dy} \right)^2 \) as:
\[ \left( \frac{dx}{dy} \right)^2 = \left( \frac{1}{\frac{dy}{dx}} \right)^2 = \frac{1}{\left( \frac{dy}{dx} \right)^2} \]

### Step 2: Substitute back into the equation
Substituting this back into the original equation gives us:
\[ y = x \left( \frac{dy}{dx} \right)^2 + \frac{1}{\left( \frac{dy}{dx} \right)^2} \]

### Step 3: Rearranging the equation
To make it easier to analyze, we can multiply through by \( \left( \frac{dy}{dx} \right)^2 \) to eliminate the fraction:
\[ y \left( \frac{dy}{dx} \right)^2 = x \left( \frac{dy}{dx} \right)^4 + 1 \]

### Step 4: Identify the highest derivative
Now we can rearrange this equation to isolate the terms involving \( \frac{dy}{dx} \):
\[ x \left( \frac{dy}{dx} \right)^4 - y \left( \frac{dy}{dx} \right)^2 + 1 = 0 \]
This equation is a polynomial in \( \left( \frac{dy}{dx} \right)^2 \).

### Step 5: Determine the order and degree
- **Order**: The order of a differential equation is the highest derivative present. Here, the highest derivative is \( \frac{dy}{dx} \), which is of order 1.
- **Degree**: The degree of a differential equation is the power of the highest derivative when the equation is a polynomial in derivatives. The highest power of \( \frac{dy}{dx} \) in our rearranged equation is 4 (from \( \left( \frac{dy}{dx} \right)^4 \)).

Thus, the degree is 4.

### Final Answer
The degree and order of the differential equation are:
\[ \text{Degree} = 4, \quad \text{Order} = 1 \]