Q.Determine the order and degree, if defined, of the differential equation: $\left(\frac{d^2y}{dx^2}\right)^2 + \cos \left(\frac{dy}{dx}\right) = 0$

The key idea is that order is the highest derivative present, and degree is the power of the highest derivative only when the equation is a polynomial in derivatives.

Step 1: The highest derivative is $\frac{d^2y}{dx^2}$, so the order is 2. …

Why this question matters

Many students rush to say "order = 2, degree = 2" because they see the square on the second derivative. But the degree of a differential equation is defined only when the equation is polynomial in all the derivatives. The moment a trigonometric, exponential, or logarithmic function wraps around any derivative, the degree becomes undefined — no matter how neat the rest looks.

Let’s walk through it carefully.

Step-by-step reasoning

1. Identify the highest-order derivative present.

The equation is:

The highest derivative appearing is $\frac{d^2y}{dx^2}$ (the second derivative). No third or higher derivative exists. So the order is $2$.

2. Check if the equation is polynomial in the derivatives.

For the degree to be defined, the differential equation must be expressible as a polynomial in all the derivatives it contains. That means:

• Every derivative term must be raised to a rational power (no fractional powers that hide radicals).
• No derivative may appear inside a trigonometric, logarithmic, exponential, or any other non-polynomial function.

Here, the term $\cos\left(\frac{dy}{dx}\right)$ has the first derivative $\frac{dy}{dx}$ trapped inside a cosine. That is not a polynomial expression — you cannot expand $\cos(u)$ as a finite sum of powers of $u$. …

Method: Order and degree when a derivative sits inside a trig function

Use this on equations like $\left(\frac{d^2y}{dx^2}\right)^2+\cos\left(\frac{dy}{dx}\right)=0$.

Steps

Step 1: Order is always readable.

The highest derivative is $\frac{d^2y}{dx^2}$, so the order is $2$ — this never depends on the polynomial test.

Step 2: Test whether the equation is polynomial in the derivatives.

A derivative trapped inside $\cos(\cdot)$ (or $\sin$, $\log$, $e^{(\cdot)}$) cannot be expanded into finitely many powers. …

Common Mistakes

Mistake 1: Answering degree $2$ from the squared second derivative.

Why it's wrong: the term $\cos\left(\frac{dy}{dx}\right)$ traps a derivative inside a cosine, so the equation is not polynomial in the derivatives and the degree is not defined — the $\left(\frac{d^2y}{dx^2}\right)^2$ does not rescue it. Correct approach: any transcendental function of a derivative makes the degree undefin …