Find the order and degree of (d²y/dx²)³ + (dy/dx)² + y = 0

Question

Find the order and degree of the differential equation:

\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + y = 0

Solution — Step by Step

Scan the equation for every derivative present:

$\dfrac{d^2y}{dx^2}$ — this is the second order derivative (differentiated twice)
$\dfrac{dy}{dx}$ — this is the first order derivative

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

Therefore, Order = 2.

For degree to be defined, the equation must be expressible as a polynomial in all its derivatives. An equation involving terms like $\sin\!\left(\dfrac{dy}{dx}\right)$ or $e^{dy/dx}$ has undefined degree.

In our equation, every derivative appears raised to an integer power — $(d^2y/dx^2)^3$ and $(dy/dx)^2$ . This is a polynomial in derivatives.

No radical signs, no transcendental functions of derivatives. Degree is well-defined.

The highest order derivative is $\dfrac{d^2y}{dx^2}$ , and it appears as:

\left(\frac{d^2y}{dx^2}\right)^3

The exponent (power) on this highest-order derivative is 3.

Therefore, Degree = 3.

Order = 2 (highest order of differentiation present)

Degree = 3 (power of the highest order derivative, after making it polynomial)

Why This Works

Order and degree measure two different things. Order tells us how many times we differentiated — what’s the “deepest” derivative. Degree tells us how “powerful” that highest derivative is within the polynomial structure of the equation.

Think of it this way: order is about the type of derivative, degree is about the exponent on that derivative.

The key restriction for degree: the equation must be written as a polynomial in its derivatives (no fractions with derivatives in denominators, no transcendental functions of derivatives). Once it is, the exponent on the highest-order derivative gives the degree.

Alternative Method — Recognise the Structure Directly

Read the equation term by term:

First term: $(d^2y/dx^2)^3$ — second-order derivative raised to power 3
Second term: $(dy/dx)^2$ — first-order derivative raised to power 2
Third term: $y$ — the function itself (zeroth derivative)

Highest order = 2 (from second-order derivative in term 1). Power of that highest-order derivative = 3.

Order = 2, Degree = 3. Done.

Common Mistake

Confusing the order with the degree. Students see the exponent 3 prominently and report “order = 3.” But order is determined by the ORDER of the derivative (how many times we differentiate), not by the power it’s raised to. The $d^2y/dx^2$ notation means “differentiate twice” — so order is 2, regardless of the cube outside. The cube only affects the degree.

In board exams, degree is sometimes undefined — whenever a derivative appears inside a square root, logarithm, sine, or exponential. If the equation is $\sqrt{d^2y/dx^2} + dy/dx = 0$ , degree is undefined (not 1) because the square root prevents polynomial form. Recognise these cases and state “degree is not defined.”

Question

Solution — Step by Step

Why This Works

Alternative Method — Recognise the Structure Directly

Common Mistake

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