The order and degree of the differential equation `y^(2) = 4a (x -a)` , where a is an arbitrary constant , are respectively

Options

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

Answer

To determine the order and degree of the given differential equation \( y^2 = 4a (x - a) \), we can follow these steps:

### Step 1: Identify the given equation
The equation provided is:
\[ y^2 = 4a (x - a) \]

### Step 2: Differentiate both sides
To analyze the order and degree, we first differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}[4a(x - a)] \]

Using the chain rule on the left side and the product rule on the right side, we get:
\[ 2y \frac{dy}{dx} = 4a \]

### Step 3: Rearrange the equation
Now, we can rearrange this equation to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{4a}{2y} = \frac{2a}{y} \]

### Step 4: Determine the order and degree
1. **Order**: The highest derivative present in the equation is \( \frac{dy}{dx} \), which is the first derivative. Therefore, the order of the differential equation is **1**.
2. **Degree**: The degree of a differential equation is determined by the power of the highest order derivative. In our case, \( \frac{dy}{dx} \) appears to the first power. Therefore, the degree of the differential equation is **1**.

### Final Answer
Thus, the order and degree of the differential equation \( y^2 = 4a (x - a) \) are:
- **Order**: 1
- **Degree**: 1

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