CBSE Sample Paper 2026-27 · Section A · 1 mark

Q11: Degree and order of $y - x\left(\frac{dy}{dx}\right)^3 = a\left(y + \frac{d^2y}{dx^2}\right)^2$

Sample 2026-27Ch 9 · Differential Equations

The question

The degree and order of the differential equation $y - x\left(\frac{dy}{dx}\right)^3 = a\left(y + \frac{d^2y}{dx^2}\right)^2$ are respectively

(A) $3, 1$

(B) $2, 2$

(D) $4, 2$

Exam answer

The derivatives in the equation are $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ ⁠.

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

Why?

The order of a differential equation is the order of the highest derivative in it. A second derivative beats a first derivative.

The equation is a polynomial in the derivatives, and the highest power of $\frac{d^2y}{dx^2}$ is 2, so the degree is 2.

Why?

The degree is the highest power of the highest order derivative, once the equation is a polynomial in its derivatives. Expanding the right side shows the power:

\begin{aligned} &\left(y + \frac{d^2y}{dx^2}\right)^2 \\ &= y^2 + 2y\frac{d^2y}{dx^2} + \left(\frac{d^2y}{dx^2}\right)^2 \end{aligned}

Watch out: the cube belongs to $\frac{dy}{dx}$ ⁠, which is not the highest order derivative, so it does not set the degree.

So the degree and the order are $2, 2$ respectively.

Why?

The question asks for the degree first, then the order. Here both are 2, so the order of writing does not matter. Option (C) takes the power 3 of $\frac{dy}{dx}$ as the degree.

Answer: (B) $2, 2$

Where this comes from

Chapter 9, Differential Equations: Order and degree of a differential equation.

Question type: Order and degree

How to spot it: click to see

Type 1. Order and degree: the highest derivative, and its power

Spot it: "find the order and degree (if defined)". Often an MCQ.

Here: The highest derivative, $\frac{d^2y}{dx^2}$ ⁠, sits inside $\left(y + \frac{d^2y}{dx^2}\right)^2$ ⁠, so it is squared: order 2, degree 2. The cube on $\frac{dy}{dx}$ does not count.

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Differential Equations in past papers: question 14 of 31 · see them all

Q11: Degree and order of y−x(dydx)3=a(y+d2ydx2)2y - x\left(\frac{dy}{dx}\right)^3 = a\left(y + \frac{d^2y}{dx^2}\right)^2y−x(dxdy​)3=a(y+dx2d2y​)2

Q11: Degree and order of $y - x\left(\frac{dy}{dx}\right)^3 = a\left(y + \frac{d^2y}{dx^2}\right)^2$