Find the differential equation of all parabolas whose axes are parallel to y-axis.

Answer

To find the differential equation of all parabolas whose axes are parallel to the y-axis, we start with the general form of such parabolas.

### Step 1: Write the general equation of the parabola
The general equation of a parabola with its axis parallel to the y-axis can be expressed as:
\[ (x - \alpha)^2 = 4a(y - \beta) \]
where \(\alpha\) and \(\beta\) are constants representing the vertex of the parabola, and \(a\) is a parameter that determines the width of the parabola.

### Step 2: Differentiate the equation with respect to \(x\)
We will differentiate the equation with respect to \(x\) to eliminate the constants. Applying implicit differentiation:
\[ 2(x - \alpha) = 4a \frac{dy}{dx} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{(x - \alpha)}{2a} \]

### Step 3: Differentiate again to eliminate \(a\)
Now we differentiate \(\frac{dy}{dx}\) with respect to \(x\):
\[ \frac{d^2y}{dx^2} = \frac{1}{2a} \]

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

### Step 5: Substitute \(a\) back into the equation
Substituting \(a\) back into the equation we derived from the first differentiation:
\[ \frac{dy}{dx} = \frac{(x - \alpha)}{2 \cdot \frac{1}{2 \frac{d^2y}{dx^2}}} = (x - \alpha) \frac{d^2y}{dx^2} \]

### Step 6: Eliminate \(\alpha\)
To eliminate \(\alpha\), we can express \(\alpha\) in terms of \(y\) and its derivatives. From the first derivative:
\[ x - \alpha = 2a \frac{dy}{dx} \]
Substituting \(a\) gives:
\[ x - \alpha = \frac{dy}{dx} \cdot \frac{1}{\frac{d^2y}{dx^2}} \implies \alpha = x - \frac{dy}{dx} \cdot \frac{1}{\frac{d^2y}{dx^2}} \]

### Step 7: Write the final differential equation
Finally, we can express the differential equation in terms of \(y\) and its derivatives:
\[ \frac{d^2y}{dx^2} = c \]
where \(c\) is an arbitrary constant.

Thus, the differential equation of all parabolas whose axes are parallel to the y-axis is:
\[ \frac{d^2y}{dx^2} = k \]
where \(k\) is a constant.