The order and degree of the differential equation `y=x(dy)/(dx)+2/("dy"/"dx")` are respectively

Options

A. 1,2
B. 1,3
C. 2,1
D. 1,1

Answer

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

### Step 1: Rewrite the Equation
Start with the given equation:
\[ y = x \frac{dy}{dx} + \frac{2}{\frac{dy}{dx}} \]

### Step 2: Multiply Both Sides by \(\frac{dy}{dx}\)
To eliminate the fraction, multiply both sides by \(\frac{dy}{dx}\):
\[ y \frac{dy}{dx} = x \left(\frac{dy}{dx}\right)^2 + 2 \]

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

### Step 4: Identify the Order and Degree
Now, we analyze the equation:
- The highest derivative present is \(\frac{dy}{dx}\), which is the first derivative. Therefore, the **order** of the differential equation is **1**.
- The degree of a differential equation is determined by the power of the highest derivative when the equation is a polynomial in derivatives. Here, \(\left(\frac{dy}{dx}\right)^2\) indicates that the highest power of the derivative is 2. Thus, the **degree** of the differential equation is **2**.

### Final Answer
The order and degree of the differential equation are respectively:
- **Order**: 1
- **Degree**: 2

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