Q.Family $y=Ax+A^3$ of curves will correspond to a differential equation of order:
(A) 3
(B) 2
(C) 1
(D) not defined

Concept: Order of a differential equation — the order of the highest derivative that appears when the arbitrary constants are eliminated.

We have one arbitrary constant $A$ in the family $y = Ax + A^3$.

Step 1: Differentiate once:

$\frac{dy}{dx} = A$.

Step 2: Substitute $A = \frac{dy}{dx}$ back into the original equation:

$y = \left(\frac{dy}{dx}\right)x + \left(\frac{dy}{dx}\right)^3$. …

The order of a differential equation is the highest order of derivative that appears in it. When we have a family of curves with one arbitrary constant, eliminating that constant typically gives a first-order differential equation. The key is to count the number of independent parameters — here, only $A$ is free.

Let’s work through the elimination step by step.

1. Start with the given family

Here $A$ is the only arbitrary constant. Differentiate once with respect to $x$:

This is immediate because the derivative of $A x$ is $A$, and $A^3$ is constant.

2. Substitute back to eliminate $A$ From the derivative, $A = \frac{dy}{dx}$. Replace $A$ in the original equation:

This is a first-order differential equation — it involves only $\frac{dy}{dx}$ and $y$, no higher derivatives.

3. Why not second order? …

Method: Order of the differential equation of a family of curves

Use this to decide the order of the equation representing a family, without necessarily solving for it.

Steps

Step 1: Count the independent arbitrary constants.

The order equals the number of genuinely independent arbitrary constants in the family. A single constant (even one raised to a power, like $A^3$) counts as one.

Step 2: Confirm by eliminating them. …

Common Mistakes

Mistake 1: Letting the cube $A^3$ inflate the order.

Why it's wrong: for $y=Ax+A^3$ students see the power $3$ and answer order $3$ (option A). But order is set by how many independent constants must be eliminated, not by any power on the constant. Correct approach: there is one constant $A$, so the order is $1$.

Mistake 2: Differentiating twice "to be safe". …