Q.Determine the order and degree, if defined, of the differential equation: The degree of the differential equation $\left(\frac{d^2 y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \sin\left(\frac{dy}{dx}\right) + 1 = 0$ is (A) 3 (B) 2 (C) 1 (D) not defined

Concept: Degree of a differential equation is defined only when the equation is a polynomial in the derivatives. If any derivative appears inside a non-polynomial function (like sine, exponential, or logarithm), the degree is not defined.

Reasoning:

1. The given equation is: $\left(\frac{d^2 y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \sin\left(\frac{dy}{dx}\right) + 1 = 0$ …

1. Recall the definition of degree.

   The degree of a differential equation is the power of the highest-order derivative, provided the equation is a polynomial in all the derivatives that appear. If the equation contains non-polynomial expressions like $\sin(y')$, $e^{y'}$, or $\log(y'')$, the degree is simply not defined.

2. Identify the highest-order derivative.

   The given equation is:

The highest-order derivative present is $\frac{d^2 y}{dx^2}$ (order 2). That part is fine — it appears as a cube, which is a polynomial term.

3. Check the problematic term.

   Look at $\sin\left(\frac{dy}{dx}\right)$. This is not a polynomial in $\frac{dy}{dx}$; it is a transcendental function of the first derivative. No amount of algebraic manipulation (squaring, cubing, etc.) can turn $\sin(y')$ into a polynomial in $y'$ — it is fundamentally non-polynomial.

4. Apply the definition strictly. …

Method: When is the degree "not defined"?

Use this whenever an equation mixes powered derivatives with a derivative inside a transcendental function.

Steps

Step 1: Find the order first (it is always defined).

The highest derivative — here $\frac{d^2y}{dx^2}$ — sets the order, independent of any polynomial test.

Step 2: Scan for a derivative inside $\sin$, $\cos$, $\log$ or $e^{(\cdot)}$.

A term like $\sin\left(\frac{dy}{dx}\right)$ cannot be rewritten as a finite polynomial in $\frac{dy}{dx}$. …

Common Mistakes

Mistake 1: Reading degree $3$ off the cubed second derivative.

Why it's wrong: the term $\sin\left(\frac{dy}{dx}\right)$ puts a derivative inside a sine, so the equation is not polynomial in the derivatives and the degree is not defined (option D) — the cube on $\frac{d^2y}{dx^2}$ is irrelevant. Correct approach: a transcendental function of a derivative always makes the degree undefined. …