Q.Which of the following is a second order differential equation?
(A) $(y')^2+x=y^2$
(B) $y'y''+y=\sin x$
(C) $y'''+(y'')^2+y=0$
(D) $y'=y^2$

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

• Option (A): $(y')^2 + x = y^2$ — highest derivative is $y'$ (first order).
• Option (B): $y' y'' + y = \sin x$ — contains $y''$ (second derivative) and no higher derivative. This is second order. …

The order of a differential equation is defined as the order of the highest derivative that appears in it. This is a definition-based question — you don’t need to solve anything, just identify which equation has $y''$ as its highest derivative and no derivative higher than that.

Let’s check each option one by one.

1. Option (A): $(y')^2 + x = y^2$

   The only derivative here is $y'$ (first derivative). There is no $y''$ or higher. So the order is 1, not 2.

2. Option (B): $y' y'' + y = \sin x$

   Here we see $y''$ (second derivative) multiplied by $y'$. The highest derivative present is $y''$, and there is no $y'''$ or higher. So the order is 2. This matches what we need.

3. Option (C): $y''' + (y'')^2 + y = 0$

   This contains $y'''$ (third derivative). Even though $y''$ also appears, the highest derivative is $y'''$, so the order is 3, not 2.

4. Option (D): $y' = y^2$

   Only $y'$ appears — order 1. …

Method: Identifying the Order of a Differential Equation

The order is simply the order of the highest derivative that appears. Powers of derivatives or products of derivatives do not change it.

Steps

Step 1: List the derivatives in each candidate.

Scan the equation for $y'$, $y''$, $y'''$, and so on.

Step 2: Pick the highest-order one.

The order equals the highest derivative present — e.g. an equation containing $y''$ but nothing higher is second order.

Step 3: Ignore powers and products when judging order. …

Common Mistakes

Mistake 1: Thinking a product like $y'y''$ raises the order.

Why it's wrong: order is the highest derivative present; $y'y''$ still has $y''$ as its highest, so the equation is second order. Correct approach: read the highest derivative and ignore products or powers.

Mistake 2: Confusing degree with order. …