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Order and degree questions look like free marks, and they are, as long as you follow the definition rather than your first impression. The traps are predictable: a square root that has to be cleared, a derivative hidden inside a sine, or an exponential that looks like it should matter but does not. This post states the two definitions exactly, then works through eight cases that cover almost every trap.
Is order and degree still on the syllabus?
Yes. The JEE (Main) 2026 syllabus lists "Ordinary differential equations, their order and degree" in Unit 9. The JEE (Advanced) 2026 brochure lists the formation of ordinary differential equations, which depends on order. The current NCERT Class 12 chapter on differential equations defines both in Sections 9.2.1 and 9.2.2, and Exercise 9.1 asks you to "determine order and degree (if defined)". Check the documents for your exam year.
What exactly are order and degree?
NCERT's definitions:
Order is "the order of the highest order derivative of the dependent variable" in the equation.
Degree is defined only when the equation is "a polynomial equation in derivatives". It is then the highest power of the highest order derivative.
Two consequences follow. Order is always defined. Degree is not always defined, and the words "if defined" in a question are a hint that one of the options may be "not defined".
NCERT also notes that an equation can be a polynomial in y′ without being a polynomial in y. Its example contains sin y and still has a degree.
Eight worked cases
Write y′ for dy/dx and y″ for d²y/dx².
(y″)³ + (y′)⁴ + y = 0. Highest derivative y″, power 3. Order 2, degree 3.
y″ + sin(y′) = 0. Not a polynomial in y′. Order 2, degree not defined.
y′ = e^(x + y). Write it as y′ = eˣ eʸ. The exponential involves y, not a derivative, so this is a polynomial in y′. Order 1, degree 1.
y‴ + e^(y′) = x. The derivative y′ is inside an exponential. Order 3, degree not defined.
[1 + (y′)²]^(3/2) = k y″. This is the radius of curvature equation. Square both sides to clear the fractional power: [1 + (y′)²]³ = k²(y″)². Order 2, degree 2.
(y″)^(2/3) = y′. Cube both sides: (y″)² = (y′)³. Order 2, degree 2.
y − x y′ = √(1 + (y′)²). Square both sides: (y − xy′)² = 1 + (y′)², which expands to (x² − 1)(y′)² − 2xy y′ + y² − 1 = 0. Order 1, degree 2.
2x² y″ − 3y′ + y = 0. This is NCERT Exercise 9.1, question 12, with options including 0 and "not defined". The highest derivative is y″, so the order is 2. The x² in front changes nothing.
How does order connect to a family of curves?
A general solution with n independent arbitrary constants usually comes from a differential equation of order n. Take y = Aeˣ + Be⁻ˣ, with two constants. Differentiate twice: y′ = Aeˣ − Be⁻ˣ and y″ = Aeˣ + Be⁻ˣ = y. Eliminating A and B gives y″ − y = 0, of order 2. This is the idea behind the "formation of differential equations" listed in the JEE Advanced syllabus.
Which mistakes cost the most marks?
Giving a degree when a derivative sits inside sin, cos, log or an exponential.
Refusing a degree because sin y or eʸ appears. Only derivatives inside such functions matter.
Reading the degree before clearing radicals and fractional powers.
Taking the highest power in the equation instead of the power of the highest derivative.
Swapping order and degree when both are small numbers. Label each answer in words every time.
How should you practise something this quick?
Because each question takes seconds, the risk is that you rely on recognition and stop thinking. Test yourself rather than rereading the rules. Roediger and Karpicke (2006) found that students who took practice tests remembered prose passages better on delayed tests than students who restudied, even though restudying made them more confident. Mixed sets help too: in a study of 126 seventh-grade students, interleaved practice beat blocked practice on tests one day and 30 days later.
Flash set (5 minutes, three times a week). Write ten equations on cards, with order and degree on the back. Answer from memory, then check.
Add traps. Make sure each set has at least two "not defined" cases and one radical to clear.
One formation question a week, like the family above.
Then move on to solving with our guide to linear differential equations.
If a definition still feels shaky, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, each with its own lesson.
Tonight, do all of NCERT Exercise 9.1 from memory, then check every answer against the two definitions above.
