Q.For each of the differential equations given below, indicate its order and degree (if defined).

Concept: Order and Degree of a Differential Equation

Order = highest derivative present. Degree = power of the highest derivative after the equation is made polynomial in derivatives (if possible).

(i) $\dfrac{d^2 y}{dx^2} + 5x \left(\dfrac{dy}{dx}\right)^2 - 6y = \log x$

Highest derivative is $\dfrac{d^2 y}{dx^2}$, which appears to the first power. No radicals or non-polynomial terms affect it.

Order = 2, Degree = 1.

(ii) $\left(\dfrac{dy}{dx}\right)^3 - 4\left(\dfrac{dy}{dx}\right)^2 + 7y = \sin x$

Highest derivative is $\dfrac{dy}{dx}$, raised to power 3.

Order = 1, Degree = 3.

(iii) $\dfrac{d^4 y}{dx^4} - \sin\left(\dfrac{d^3 y}{dx^3}\right) = 0$

Highest derivative is $\dfrac{d^4 y}{dx^4}$, but the term $\sin\left(\dfrac{d^3 y}{dx^3}\right)$ is not a polynomial in the derivative — degree is not defined.

Order = 4, Degree = not defined.

The two numbers — order and degree — are the simplest descriptors of a differential equation. Order is straightforward: it’s just the highest derivative that appears. Degree is trickier: it’s the exponent of that highest derivative after you’ve rewritten the equation so that all derivatives are raised to positive integer powers and no derivative is inside a transcendental function (like sin, cos, log, exp). If you can’t do that, degree is not defined.

Let’s apply this to each equation.

1. Equation (i): $\dfrac{d^2 y}{dx^2} + 5x \left(\dfrac{dy}{dx}\right)^2 - 6y = \log x$

   The highest derivative is $\dfrac{d^2 y}{dx^2}$ — that’s order 2.

   Now look at the equation: it’s already a polynomial in the derivatives. The term $\dfrac{d^2 y}{dx^2}$ appears with exponent 1. There’s no sine, no log of a derivative, no fractional power. So the degree is simply 1.

   Tip

   The $\log x$ on the right is a function of $x$ alone, not of $y$ or any derivative — it doesn’t affect the degree at all. Only derivatives matter.

2. Equation (ii): $\left(\dfrac{dy}{dx}\right)^3 - 4\left(\dfrac{dy}{dx}\right)^2 + 7y = \sin x$

   The highest derivative is $\dfrac{dy}{dx}$ — order 1.

   The equation is already a polynomial in that derivative: the term $\left(\dfrac{dy}{dx}\right)^3$ has exponent 3. No further manipulation is needed. So degree is 3.

   Watch out

   A common mistake is to think the degree is the highest power among all terms — here someone might say 3 because of the cube, but that’s actually correct in this case. The real pitfall is when the highest derivative itself has a fractional power or is inside a function; then you must rationalise first.

3. Equation (iii): $\dfrac{d^4 y}{dx^4} - \sin\left(\dfrac{d^3 y}{dx^3}\right) = 0$

   The highest derivative is $\dfrac{d^4 y}{dx^4}$ — order 4.

   Now for degree: the equation contains $\sin\left(\dfrac{d^3 y}{dx^3}\right)$. That’s a transcendental function of a derivative. You cannot expand $\sin(u)$ as a finite polynomial in $u$ — it’s an infinite series. So the equation cannot be written as a polynomial in the derivatives. Hence degree is not defined.

   Important

   Whenever a derivative appears inside a trigonometric, logarithmic, exponential, or any non-polynomial function, the degree is not defined — regardless of the order.

Method: Reading off order and degree

For any differential equation, order and degree are found by inspection using two clear rules.

Steps

Step 1: Find the order.

The order is the highest-order derivative that appears (the one with the most differentiations).

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

Degree is defined only if every derivative appears with a whole-number power and no derivative sits inside a function like $\sin,\ \log,\ e^{(\cdot)}$ or under a root.

Step 3: Read the degree.

If it is polynomial in the derivatives, the degree is the power of the highest-order derivative.

Step 4: Declare 'not defined' when needed.

If a derivative is wrapped in a transcendental function (e.g. $\sin\!\big(\tfrac{d^3y}{dx^3}\big)$), the degree is not defined — never expand it into a series.

Common Mistakes

Mistake 1: Taking the highest power of any derivative as the degree.

Why it's wrong: in (i), $\left(\tfrac{dy}{dx}\right)^2$ appears, but the degree is the power of the highest-order derivative $\tfrac{d^2y}{dx^2}$, which is $1$. Correct approach: degree looks only at the top-order derivative.

Mistake 2: Assigning a degree to (iii).

Why it's wrong: $\sin\!\left(\tfrac{d^3y}{dx^3}\right)$ wraps a derivative in a transcendental function, so the equation is not polynomial in derivatives and the degree is not defined. Correct approach: state "degree not defined," never expand the sine.

Mistake 3: Miscounting the order.

Why it's wrong: the order is the highest derivative present — $4$ in (iii), not $3$. Correct approach: scan for the largest number of differentiations.