Find the differential equation of all parabolas whose axes are parallel to the x-axis an having latus rectum a.

Explanation

To find the differential equation of all parabolas whose axes are parallel to the x-axis and having a latus rectum of length \( a \), we can follow these steps:

### Step 1: Write the equation of the parabola
The standard form of a parabola that opens horizontally (with its axis parallel to the x-axis) is given by:
\[ y^2 = 4ax \]
where \( a \) is the distance from the vertex to the focus, and also half the length of the latus rectum.

### Step 2: Introduce parameters
Since we are considering a family of parabolas, we can introduce parameters \( \alpha \) and \( \beta \) to represent the vertex of the parabola. The equation can be modified to:
\[ (y - \beta)^2 = 4a(x - \alpha) \]
This represents a family of parabolas with vertex at \( (\alpha, \beta) \).

### Step 3: Differentiate the equation
To find the differential equation, we need to eliminate the parameters \( \alpha \) and \( \beta \). First, we differentiate the equation with respect to \( x \):
\[ 2(y - \beta) \frac{dy}{dx} = 4a \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{2(y - \beta)}{a} \]

### Step 4: Differentiate again
Next, we differentiate the expression for \( \frac{dy}{dx} \) with respect to \( x \) to find the second derivative:
\[ \frac{d^2y}{dx^2} = \frac{2}{a} \left( \frac{dy}{dx} \right) \frac{d}{dx}(y - \beta) \]
Since \( \beta \) is a constant, we have:
\[ \frac{d^2y}{dx^2} = \frac{2}{a} \frac{dy}{dx} \]

### Step 5: Substitute back to eliminate parameters
From the first derivative, we have:
\[ y - \beta = \frac{a}{2} \frac{dy}{dx} \]
Substituting this back into the second derivative equation gives:
\[ \frac{d^2y}{dx^2} = \frac{2}{a} \frac{dy}{dx} \cdot \frac{a}{2} \frac{dy}{dx} = \frac{1}{a} \left( \frac{dy}{dx} \right)^2 \]

### Step 6: Form the differential equation
Rearranging gives us the required differential equation:
\[ a \frac{d^2y}{dx^2} + \left( \frac{dy}{dx} \right)^2 = 0 \]

### Final Result
Thus, the differential equation of all parabolas whose axes are parallel to the x-axis and having a latus rectum of length \( a \) is:
\[ a \frac{d^2y}{dx^2} + \left( \frac{dy}{dx} \right)^2 = 0 \]