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.