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

Answer

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: Identify the Differential Equation
The given equation is:
\[ y = x \frac{dy}{dx} + \left(\frac{dy}{dx}\right)^{-1} \]

### Step 2: Multiply the Entire Equation by \(\frac{dy}{dx}\)
To eliminate the inverse term, we multiply both sides of the equation by \(\frac{dy}{dx}\):
\[ y \frac{dy}{dx} = x \left(\frac{dy}{dx}\right)^2 + 1 \]

### Step 3: Rearrange the Equation
Rearranging the equation gives us:
\[ x \left(\frac{dy}{dx}\right)^2 - y \frac{dy}{dx} + 1 = 0 \]

### Step 4: Identify the Highest Power of the Derivative
In the rearranged equation, we can see that the highest power of \(\frac{dy}{dx}\) is 2 (it appears as \(\left(\frac{dy}{dx}\right)^2\)).

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

Thus, the degree of the differential equation \( y = x \frac{dy}{dx} + \left(\frac{dy}{dx}\right)^{-1} \) is 2.

### Final Answer
The degree of the differential equation is 2.

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