Q.Determine the order and degree, if defined, of the differential equation: $\frac{d^4y}{dx^4} + \sin(y''') = 0$

The key idea is that degree is defined only when the differential equation is polynomial in the derivatives. Here, the term $\sin(y''')$ is non-polynomial.

Step 1: Identify the highest order derivative.

The term $\frac{d^4y}{dx^4}$ gives the highest derivative, so the order is 4.

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

The equation contains $\sin(y''')$, which is a transcendental function of the third derivative. This cannot be expressed as a polynomial in $y'''$ (or any derivative).

Step 3: Conclude about the degree.

Since the equation is not a polynomial in the derivatives, the degree is not defined.

Why this question matters

Many students rush to count derivatives and then mechanically look for an exponent. But the degree of a differential equation is defined only when the equation is a polynomial in all the derivatives that appear. If any derivative is inside a trigonometric, exponential, logarithmic, or other non-polynomial function, the degree simply does not exist — no matter how tidy the rest looks.

Here, the equation is:

where $y'''$ means $\frac{d^3y}{dx^3}$.

Step-by-step reasoning

1. Identify the order.

   The order is the highest derivative present. We see $\frac{d^4y}{dx^4}$ (the fourth derivative) and $y'''$ (the third derivative). The highest is 4, so the order is 4.

2. Check if degree is defined.

   Degree is defined only when the differential equation can be written as a polynomial in the derivatives, with all exponents being non‑negative integers. Look at the term $\sin(y''')$ — it is a sine of a derivative. No algebraic manipulation can turn $\sin(y''')$ into a polynomial in $y'''$ (or any other derivative). The sine function is transcendental, not algebraic.

3. Conclude about degree.

   Because the equation contains a non‑polynomial function of a derivative, the degree is not defined. This is a standard exam point: if you see $\sin(y')$, $\cos(y'')$, $e^{y'}$, $\log(y')$, etc., the degree is undefined — even if the rest of the equation looks polynomial.

Method: Determining the order and degree of a differential equation

This is the standard classification technique for any differential equation, before you decide how to solve it.

Steps

Step 1: Find the order

The order is the order of the highest derivative that appears. Scan for $\dfrac{d^ny}{dx^n}$ (or $y^{(n)}$) with the largest $n$ — that $n$ is the order, regardless of any powers or functions wrapped around it.

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

Degree is defined only when the equation can be written as a polynomial in all the derivatives (after clearing radicals and fractions). If any derivative sits inside a transcendental function — $\sin(\cdot)$, $\cos(\cdot)$, $e^{(\cdot)}$, $\log(\cdot)$ — the equation is not polynomial in the derivatives.

Step 3: State the degree (or that it is undefined)

If it is polynomial, the degree is the power of the highest-order derivative. If a derivative is trapped inside a transcendental function, the degree is not defined.

Common Mistakes

Mistake 1: Assigning a degree even though $\sin(y''')$ is present

Why it's wrong: degree is defined only for equations polynomial in the derivatives; a derivative inside $\sin(\cdot)$ is transcendental. Correct approach: state the degree is not defined.

Mistake 2: Confusing order with degree

Why it's wrong: the order (highest derivative $=4$ here) is unaffected by the $\sin$ wrapper — only the degree becomes undefined. Correct approach: report order $=4$, degree undefined.