Find the differential equation of all non-vertical lines in a plane.

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Q. Find the differential equation of all non-vertical lines in a plane.

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Concept understanding — Eliminating Arbitrary Constant

Eliminating Arbitrary Constants – The Core Idea

Consider a family of curves — say all circles centred at the origin: $x^2 + y^2 = r^2$ . Here $r$ is an arbitrary constant: each value gives one specific circle, but the whole family shares the same shape.

Suppose you're asked: "What differential equation does every member satisfy?" You need to eliminate the arbitrary constant $r$ to get an equation that holds for all such circles, regardless of $r$ .

That is the goal: start with an equation containing arbitrary constants, differentiate enough times to remove them, and obtain a differential equation representing the whole family.

Why Differentiate?

Arbitrary constants are constants — their derivative is zero. So differentiating either makes the constant disappear or lets you eliminate it by combining the original equation with its derivatives.

Differentiate $x^2 + y^2 = r^2$ with respect to $x$ : $\;2x + 2y \frac{dy}{dx} = 0$ . Divide by 2:

x + y \frac{dy}{dx} = 0

The constant $r$ is gone. The differential equation $x + y y' = 0$ is satisfied by every circle centred at the origin.

Note

We needed one differentiation because there was one arbitrary constant. In general, $n$ arbitrary constants require $n$ differentiations.

The Precise Statement

Eliminating arbitrary constants means: given an equation involving $x$ , $y$ , and $n$ arbitrary constants, differentiate it $n$ times (with respect to the independent variable) and then algebraically eliminate the $n$ constants from the system of $n+1$ equations (the original plus the $n$ derivatives). The result is an $n$ -th order ordinary differential equation representing the entire family of curves.

A Second Example (Two Constants)

Take all curves $y = A x + B x^2$ , where $A$ and $B$ are arbitrary constants.

Differentiate once: $\;y' = A + 2B x$

Differentiate again: $\;y'' = 2B$

Now we have three equations:

$y = A x + B x^2$
$y' = A + 2B x$
$y'' = 2B$

From (3), $B = \frac{y''}{2}$ . Substitute into (2): $y' = A + x y''$ , so $A = y' - x y''$ .

Substitute $A$ and $B$ into (1):

y = (y' - x y'') x + \left(\frac{y''}{2}\right) x^2 = x y' - \frac{x^2 y''}{2}

Multiply through by 2 and rearrange:

x^2 y'' - 2x y' + 2y = 0

This second-order ODE is satisfied by every curve $y = A x + B x^2$ , no matter what $A$ and $B$ are.

Watch out

A common mistake: trying to eliminate constants by substitution before differentiating enough times. You must differentiate first, then eliminate — substituting too early loses information.

Why This Matters

In physics and engineering you often know the family of possible solutions (e.g. all motions of a spring) but not the specific constants (initial conditions). The differential equation — free of arbitrary constants — describes the law governing all members of the family: it takes you from "a set of possible curves" to "the rule they all obey."

Quick Summary

Number of arbitrary constants	Differentiations needed	Resulting ODE order
1	1	1st order
2	2	2nd order
$n$	$n$	$n$ th order

Important

The final differential equation must contain no arbitrary constants — only $x$ , $y$ , and derivatives of $y$ . If any constant remains, you haven't eliminated them all.

Forming a differential equation by eliminating arbitrary constants is one of the earliest and most tested skills in the NCERT Class 12 Differential Equations chapter, appearing in nearly every CBSE board paper and JEE Main sitting. "Formation of differential equation by eliminating arbitrary constants examples" is a top search term, since the number of constants directly determines how many times you must differentiate.

The key idea is that a non-vertical line has exactly one parameter (its slope) after fixing the intercept, so the differential equation must be second-order to eliminate two arbitrary constants. The result is $\frac{d^2 y}{dx^2} = 0$ .

The problem asks for the differential equation satisfied by every non-vertical straight line in the plane. That means we start with the general equation of such a line, which contains arbitrary constants, and then differentiate to eliminate those constants. The result is a relation involving only derivatives — the differential equation.

Why does this work? A differential equation is essentially a constraint on the rate of change of a function. If a family of curves (here, all non-vertical lines) shares a common geometric property, that property translates into a condition on derivatives. For a straight line, the defining property is constant slope. That constant slope is the first derivative, but it varies from line to line — so it's still an arbitrary constant. To get rid of both constants (the slope and the intercept), we need to go one step further: the second derivative.

Let’s do it step by step.

Write the general equation of a non-vertical line. Any non-vertical line can be written as

y = mx + c

where $m$ is the slope and $c$ is the $y$ -intercept. Both $m$ and $c$ are arbitrary constants (they can be any real numbers). The condition "non-vertical" simply means $m$ is finite — we don't need to worry about vertical lines ( $x = \text{constant}$ ) because their slope is undefined.

Differentiate once with respect to $x$ . Since $y$ is a function of $x$ ,

\frac{dy}{dx} = m

The slope $m$ is still present. So one differentiation hasn't eliminated all arbitrary constants — $m$ remains.

Differentiate a second time. Differentiate $\frac{dy}{dx} = m$ with respect to $x$ :

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

The constant $m$ disappears because the derivative of a constant is zero. The constant $c$ also vanished after the first differentiation. So now we have an equation involving only the second derivative — no arbitrary constants left.

Interpret the result. The equation $\frac{d^2 y}{dx^2} = 0$ says: the rate of change of the slope is zero. That is, the slope is constant. This is exactly the geometric property of a straight line. Every non-vertical line satisfies this, and conversely, any function whose second derivative is zero is a straight line (since integrating twice gives $y = Ax + B$ ).

Watch out

A common mistake is to stop at $\frac{dy}{dx} = m$ and call that the differential equation. But that still contains the arbitrary constant $m$ , so it's not a differential equation of the family — it's just the derivative of a particular line. The differential equation must be free of all arbitrary constants.

Tip

Notice that the order of the differential equation equals the number of arbitrary constants in the general equation. Here we had two constants ( $m$ and $c$ ), so we needed a second-order equation. This is a useful rule of thumb: to eliminate $n$ independent arbitrary constants, you generally need an $n$ th-order differential equation.

✓Final answer

\boxed{\frac{d^2 y}{dx^2} = 0}

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Q.Find the differential equation of all non-vertical lines in a plane.

Concept understanding — Eliminating Arbitrary Constant

Eliminating Arbitrary Constants – The Core Idea

Why Differentiate?

The Precise Statement

A Second Example (Two Constants)

Why This Matters

Quick Summary

Method: Forming a DE by eliminating arbitrary constants

Steps

Common Mistakes

Q.Find the differential equation of all non-vertical lines in a plane.

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