Q.The order of the differential equation $2x^2 \dfrac{d^2 y}{dx^2} - 3 \dfrac{dy}{dx} + y = 0$ is (A) 2 (B) 1 (C) 0 (D) not defined

The key idea is that the order of a differential equation is the highest order derivative present in the equation.

Step 1: Identify all derivatives of $y$ with respect to $x$ in the given equation:

$2x^2 \frac{d^2 y}{dx^2} - 3 \frac{dy}{dx} + y = 0$ …

The order of a differential equation is one of the first things you check — it tells you the highest derivative that appears in the equation. Think of it like the "depth" of the equation: a first-order equation involves only the first derivative, a second-order involves the second derivative, and so on. No tricks, no hidden conditions — just look for the derivative with the largest exponent of differentiation.

Let’s apply that directly to the given equation.

1. Identify all derivatives present. The equation is

The terms are:

• $\frac{d^2 y}{dx^2}$ — the second derivative of $y$ with respect to $x$.
• $\frac{dy}{dx}$ — the first derivative.
• $y$ — the function itself (zeroth derivative).

2. Find the highest order derivative.

   Among these, the derivative with the highest order is $\frac{d^2 y}{dx^2}$, which is of order 2. No derivative of order 3 or higher appears.

3. Check for any hidden conditions. …

Method: Reading off the order of a differential equation

Use this for a bare "what is the order?" question, such as for $2x^2\frac{d^2y}{dx^2}-3\frac{dy}{dx}+y=0$.

Steps

Step 1: List every derivative that appears.

Here: $\frac{d^2y}{dx^2}$, $\frac{dy}{dx}$, and $y$ (the zeroth derivative).

Step 2: The order is the highest of these.

$\frac{d^2y}{dx^2}$ is the highest, so the order is $2$. …

Common Mistakes

Mistake 1: Letting the coefficient $2x^2$ suggest a higher order.

Why it's wrong: variable coefficients do not change the order; the highest derivative $\frac{d^2y}{dx^2}$ fixes the order at $2$. Correct approach: ignore coefficients when reading the order.

Mistake 2: Reporting the degree instead of the order. …