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

Step 1: The given equation is

$\left(\frac{d^2y}{dx^2}\right)^2+\left(\frac{dy}{dx}\right)^2=x\sin\left(\frac{dy}{dx}\right).$ …

The degree of a differential equation is a surprisingly subtle idea. Many students rush to look at the highest power of the highest derivative — here they see $\left(\frac{d^2y}{dx^2}\right)^2$ and immediately say "degree 2". But that's only half the story.

The key condition: For a differential equation to have a degree, it must be expressible as a polynomial in the derivatives (with the dependent variable and its derivatives as the only variables). No trigonometric functions, no exponentials, no logarithms of derivatives — those break the polynomial form.

Let's check our equation step by step.

1. Write the equation clearly

2. Identify the derivatives present

   • First derivative: $\frac{dy}{dx}$
   • Second derivative: $\frac{d^2y}{dx^2}$ The equation involves both.

3. Check if it is a polynomial in derivatives

   The left-hand side is fine: $\left(\frac{d^2y}{dx^2}\right)^2$ and $\left(\frac{dy}{dx}\right)^2$ are polynomial terms (powers of derivatives).

   But the right-hand side contains $\sin\left(\frac{dy}{dx}\right)$. This is a trigonometric function of a derivative, not a polynomial in it. You cannot expand $\sin(y')$ as a finite sum of powers of $y'$ — it's transcendental. …

Method: Deciding the Degree of a Differential Equation

Degree is the power of the highest-order derivative after the equation has been made a polynomial in all its derivatives. The first job is to test whether that is even possible.

Steps

Step 1: Scan for any derivative inside a non-polynomial function.

Look for a derivative trapped inside $\sin$, $\cos$, $\log$, $e^{(\cdot)}$, or a fractional/irrational power.

Step 2: If such a term exists, the degree is not defined.

An equation that cannot be written as a polynomial in its derivatives has no degree — for example any term like $\sin\!\left(\frac{dy}{dx}\right)$ ends the analysis immediately. …

Common Mistakes

Mistake 1: Reading the degree as $2$ from the $\left(\frac{d^2y}{dx^2}\right)^2$ term.

Why it's wrong: degree is only meaningful once the whole equation is a polynomial in its derivatives; the $\sin\!\left(\frac{dy}{dx}\right)$ term breaks that condition. Correct approach: check every term for a derivative inside a transcendental function before assigning any degree.

Mistake 2: Confusing "not defined" with "zero" or "infinite". …