The differential equation of all parabolas whose axis of symmetry is along X-axis is of order.

Options

A. 2
B. 3
C. 1
D. None of these

Answer

To find the order of the differential equation of all parabolas whose axis of symmetry is along the X-axis, we can follow these steps:

### Step 1: Write the standard form of the parabola
The standard equation of a parabola with its axis of symmetry along the X-axis can be expressed as:
\[ y - k = a(x - h)^2 \]
where \( (h, k) \) is the vertex of the parabola.

### Step 2: Differentiate the equation
To find the differential equation, we first differentiate the equation with respect to \( x \):
\[ \frac{dy}{dx} = 2a(x - h) \]

### Step 3: Differentiate again
Next, we differentiate again to find the second derivative:
\[ \frac{d^2y}{dx^2} = 2a \]

### Step 4: Express \( a \) in terms of derivatives
From the second derivative, we can express \( a \) as:
\[ a = \frac{1}{2} \frac{d^2y}{dx^2} \]

### Step 5: Substitute \( a \) back into the equation
Substituting \( a \) back into the first derivative equation, we have:
\[ \frac{dy}{dx} = \frac{1}{2} \frac{d^2y}{dx^2} (x - h)^2 \]

### Step 6: Eliminate \( h \)
To eliminate \( h \), we can express \( h \) in terms of \( y \) and \( x \). However, for the purpose of finding the order of the differential equation, we will focus on the derivatives we have obtained.

### Step 7: Form the differential equation
Now, we can form a relationship involving the derivatives:
\[ \frac{d^2y}{dx^2} = 2a \]
Substituting \( a \) gives us a relationship involving the first and second derivatives.

### Step 8: Differentiate to find the third derivative
Differentiating the second derivative gives us:
\[ \frac{d^3y}{dx^3} = 0 \]
This indicates that the third derivative is constant, which is a characteristic of parabolas.

### Conclusion: Determine the order
The highest derivative that appears in our final expression is the third derivative, which means the order of the differential equation is:
\[ \text{Order} = 3 \]