What is the order of the differential equation ?
` (dy)/(dx)+y=1/((dy)/(dx))`

Options

A. -1
B. 0
C. 1
D. 2

Explanation

To determine the order of the given differential equation \( \frac{dy}{dx} + y = \frac{1}{\frac{dy}{dx}} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ \frac{dy}{dx} + y = \frac{1}{\frac{dy}{dx}} \]
Let’s denote \( \frac{dy}{dx} \) as \( y' \) for simplicity. Thus, we can rewrite the equation as:
\[ y' + y = \frac{1}{y'} \]

### Step 2: Multiply both sides by \( y' \)
To eliminate the fraction, we multiply both sides by \( y' \):
\[ y' \cdot (y' + y) = 1 \]
This simplifies to:
\[ (y')^2 + y \cdot y' = 1 \]

### Step 3: Rearrange the equation
We can rearrange the equation to make it clearer:
\[ (y')^2 + y y' - 1 = 0 \]

### Step 4: Identify the highest derivative
In the equation \( (y')^2 + y y' - 1 = 0 \), the highest derivative present is \( y' \) (which is \( \frac{dy}{dx} \)). The highest power of the derivative \( y' \) is 2.

### Step 5: Determine the order
The order of a differential equation is defined as the highest derivative present in the equation. Since the highest power of \( y' \) is 2, the order of the differential equation is:
\[ \text{Order} = 2 \]

### Final Answer
Thus, the order of the differential equation \( \frac{dy}{dx} + y = \frac{1}{\frac{dy}{dx}} \) is **2**.

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