Q.(xi) The differential equation of all non-horizontal lines in a plane is $\frac{d^2x}{dy^2}=0$. (State True or False.)

Concept: Order of a differential equation — the order of the highest derivative present.

A non-horizontal line in the $xy$-plane has the form $y = mx + c$, where $m \neq 0$ (if $m = 0$, the line is horizontal).

1. Differentiate with respect to $x$: $\frac{dy}{dx} = m$.
2. Differentiate again: $\frac{d^2y}{dx^2} = 0$. This is the standard second-order DE for all non-vertical lines. …

The question asks you to judge a statement: "The differential equation of all non-horizontal lines in a plane is $\frac{d^2x}{dy^2}=0$." This is a True/False problem, but it tests a deeper understanding of what a differential equation represents and how we derive it from a family of curves.

Let's break it down.

1. What does "non-horizontal lines" mean?

   A horizontal line has the form $y = c$ (constant slope zero). A non-horizontal line is any line that is not parallel to the x-axis — so it can be written as $y = mx + c$ with $m \neq 0$, or equivalently as $x = py + q$ (where $p \neq 0$). The second form is more useful here because the given equation uses derivatives with respect to $y$.

2. Why consider $x$ as a function of $y$?

   Usually we write $y = f(x)$, but for a vertical line ($x = \text{constant}$) the slope $\frac{dy}{dx}$ is undefined. However, the problem specifically says non-horizontal lines — vertical lines are allowed. To include vertical lines, we treat $x$ as a function of $y$. A non-horizontal line can always be written as $x = ay + b$, where $a, b \in \mathbb{R}$ are constants — this includes $a=0$, which gives the vertical line $x=b$ (still non-horizontal). So the family is $x = ay + b$, with $a, b \in \mathbb{R}$.

3. Derive the differential equation.

   Differentiate $x = ay + b$ with respect to $y$:

Differentiate again:

This is a second-order differential equation. It has no arbitrary constants left — we eliminated both $a$ and $b$ by differentiating twice. So every non-horizontal line satisfies $\frac{d^2x}{dy^2}=0$.

4. But is the converse true? …

Method: Forming the DE of a Family of Straight Lines

To find the differential equation of a family of lines, write the family with its arbitrary constants, then differentiate enough times to eliminate every constant. The number of constants sets the order.

Steps

Step 1: Write the family so the constants are explicit.

Choose the description that captures the whole family — for "non-horizontal" lines, writing $x$ as a function of $y$, i.e. $x=py+q$, is natural because it also includes vertical lines.

Step 2: Differentiate to remove the constants. …

Common Mistakes

Mistake 1: Writing $\frac{d^2y}{dx^2}=0$ instead of $\frac{d^2x}{dy^2}=0$.

Why it's wrong: $\frac{d^2y}{dx^2}=0$ describes non-vertical lines $y=mx+c$ and excludes vertical ones; to include vertical lines you treat $x$ as a function of $y$. Correct approach: for all non-horizontal lines use $x=py+q$, giving $\frac{d^2x}{dy^2}=0$.

Mistake 2: Guessing the order without counting constants. …