What is the degree of the differential equation
` y = x (dy)/(dx) + ((dy)/(dx))^(-1)` ?

Options

A. 1
B. 2
C. -1
D. Degree does not exist

Explanation

To find the degree of the given differential equation \( y = x \frac{dy}{dx} + \left( \frac{dy}{dx} \right)^{-1} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ y = x \frac{dy}{dx} + \left( \frac{dy}{dx} \right)^{-1} \]

### Step 2: Multiply through by \( \frac{dy}{dx} \)
To eliminate the negative exponent, we multiply both sides by \( \frac{dy}{dx} \):
\[ y \frac{dy}{dx} = x \left( \frac{dy}{dx} \right)^2 + 1 \]

### Step 3: Rearrange the equation
Now we can rearrange the equation to isolate all terms on one side:
\[ x \left( \frac{dy}{dx} \right)^2 - y \frac{dy}{dx} + 1 = 0 \]

### Step 4: Identify the highest power of \( \frac{dy}{dx} \)
In the rearranged equation, we see that the highest power of \( \frac{dy}{dx} \) is 2 (from the term \( x \left( \frac{dy}{dx} \right)^2 \)).

### Step 5: Determine the degree
The degree of a differential equation is defined as the highest power of the highest order derivative present in the equation. Since the highest power of \( \frac{dy}{dx} \) is 2, we conclude that the degree of the differential equation is:
\[ \text{Degree} = 2 \]

### Final Answer
The degree of the differential equation \( y = x \frac{dy}{dx} + \left( \frac{dy}{dx} \right)^{-1} \) is **2**.

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