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The problem asks for the differential equation corresponding to a family of parallel lines. Let's break this down.

### Step 1: Identify the general form of the lines
The family of parallel lines can be represented in the form:

\[
y = mx + c
\]

Since the lines are parallel, they all have the same slope, \(m\), but different intercepts, \(c\).

### Step 2: Implicit form of the equation
The equation \(y = mx + c\) can be rearranged into its implicit form:

\[
y - mx - c = 0
\]

### Step 3: Derive the differential equation
Since \(c\) is the parameter that varies to give different parallel lines, we eliminate it to find the differential equation. Differentiating both sides of the implicit form with respect to \(x\):

\[
\frac{d}{dx}(y - mx - c) = \frac{dy}{dx} - m = 0
\]

This simplifies to:

\[
\frac{dy}{dx} = m
\]

Since \(m\) is constant (the slope of the parallel lines), and there is no dependence on \(x\) or \(y\), the equation for \(m\) should actually involve differentiating again to eliminate \(c\). Differentiate \(\frac{dy}{dx} = m\) again:

\[
\frac{d^2y}{dx^2} = 0
\]

### Conclusion
The differential equation representing the family of parallel lines is:

\[
\frac{d^2y}{dx^2} = 0
\]

This equation indicates that the slope of the tangent (i.e., \(\frac{dy}{dx}\)) is constant, which is characteristic of a family of parallel lines.

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Would you like more details on this solution or have any questions?

### Follow-up Questions:
1. What is the meaning of the differential equation \(\frac{d^2y}{dx^2} = 0\) in terms of geometry?
2. How can we interpret the first derivative \(\frac{dy}{dx}\) for a family of lines?
3. What would the differential equation look like for a family of concentric circles?
4. How would the solution differ if the lines were not parallel?
5. Can you explain how the elimination of the constant \(c\) leads to the final differential equation?

### Tip:
Always remember that the order of the differential equation corresponds to the number of independent parameters (e.g., slopes, intercepts) that are eliminated during its derivation.

Explore the differential equation \(\frac{d^2y}{dx^2} = 0\) and its representation of parallel lines. This problem delves into differential equations, focusing on constant slope and parallel line characteristics. Suitable for advanced high school students and enthusiasts of mathematical analysis.

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