The degree and order of the differential equation of the family of all parabolas whose axis is x-axs are respectively

Options

A. 2,1
B. 1,2
C. 3,2
D. none of these

Explanation

To find the degree and order of the differential equation of the family of all parabolas whose axis is the x-axis, we can follow these steps:

### Step 1: Write the General Equation of the Parabola
The general equation of a parabola with its axis along the x-axis can be written as:
\[ y^2 = 4ax - t \]
where \( a \) and \( t \) are arbitrary constants.

### Step 2: Differentiate the Equation
Now, we differentiate both sides of the equation with respect to \( x \):
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4ax - t) \]

Using the chain rule on the left side and the fact that \( a \) and \( t \) are constants:
\[ 2y \frac{dy}{dx} = 4a \cdot 1 - 0 \]
This simplifies to:
\[ 2y \frac{dy}{dx} = 4a \]

### Step 3: Solve for \( a \)
Rearranging the equation gives:
\[ y \frac{dy}{dx} = 2a \]
From this, we can express \( a \) in terms of \( y \) and \( \frac{dy}{dx} \):
\[ a = \frac{y \frac{dy}{dx}}{2} \]

### Step 4: Differentiate Again
Since we have two arbitrary constants \( a \) and \( t \), we need to differentiate the equation again. We differentiate \( y \frac{dy}{dx} = 2a \):
Using the product rule:
\[ \frac{d}{dx}(y \frac{dy}{dx}) = \frac{d}{dx}(2a) \]

This gives:
\[ \frac{dy}{dx} \cdot \frac{dy}{dx} + y \cdot \frac{d^2y}{dx^2} = 0 \]
or:
\[ \left(\frac{dy}{dx}\right)^2 + y \frac{d^2y}{dx^2} = 0 \]

### Step 5: Identify the Order and Degree
Now, we analyze the resulting differential equation:
\[ y \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 0 \]

- **Order**: The order of the differential equation is the highest derivative present. Here, the highest derivative is \( \frac{d^2y}{dx^2} \), so the order is **2**.
- **Degree**: The degree of a differential equation is the power of the highest order derivative when the equation is a polynomial in derivatives. In this case, \( \frac{d^2y}{dx^2} \) is to the power of 1, and \( \left(\frac{dy}{dx}\right)^2 \) is to the power of 2, but since we are considering the highest order term, the degree is **1**.

### Final Answer
Thus, the degree and order of the differential equation of the family of all parabolas whose axis is the x-axis are:
- **Order**: 2
- **Degree**: 1