Order and degree, and the polynomial condition degree needs before it means anything

What to assume they know

• Higher derivatives, and what taking a derivative twice or three times means
• The three notations for a derivative fixed in the previous topic
• What makes an expression a polynomial in a stated symbol
• Powers with whole-number exponents, and reading an exponent off a bracket
• The sine function applied to an arbitrary argument
• Distinguishing an argument of a function from a factor multiplying it

What they should be able to do

• Find the highest derivative in a given equation and state the order
• Explain why a large power on a low derivative does not raise the order
• State the condition that must hold before degree is defined, and apply it
• Decide whether a given equation is a polynomial in its derivatives, ignoring how the dependent variable itself appears
• Give the degree, when it exists, as the power attached to the derivative that fixed the order
• Say "not defined" for degree, with a reason, when the condition fails
• Recognise the two shapes that most often destroy the condition: a derivative inside a trigonometric function, and a derivative inside an exponential
• State that order and degree are both whole numbers greater than zero
• Work an item of the chapter's own exercise from start to finish, giving both labels and defending each

Where it usually goes wrong

• "Degree is the biggest exponent in the equation." It is the exponent on the derivative that fixed the order, and nothing else. Equation eight on Part II p. 302 exists to punish this reading, and Example 1 part two repeats it a page later.
• "If there is a square root anywhere, the degree is a half." The chapter's boxed note forecloses this: degree, where it exists, is a whole number above zero. An equation carrying a fractional power of a derivative must be cleared of the root before the question is even asked, and if it cannot be cleared the answer is that degree is not defined.
• "The equation must be a polynomial, full stop." It must be a polynomial in the derivatives. The chapter's tenth equation carries the square of the sine of y and is still granted a degree, and the chapter flags the point in a parenthesis so that nobody generalises.
• "A sine anywhere kills the degree." Only a derivative inside the sine does. Exercise 9.1 items four and ten are the controlled comparison: a derivative inside the cosine in one, y inside the sine in the other, and only the first loses its degree.
• "Not defined means zero." It means the question does not arise. Writing zero asserts a whole number the boxed note has already excluded.
• "Order can be zero if no derivative appears." An equation with no derivative is not one of these equations at all, which is the whole content of the previous topic.
• "I should simplify first and then read the labels." Rearranging is fine and often necessary; but a student who divides by a derivative, or takes a root of both sides to make an equation look tidier, can change the answer. Do algebra that clears roots and denominators, and stop there.
• "Order tells me how hard the equation is." It tells you how many times something was differentiated. The chapter's own hardest worked examples are all of order one.

Questions to check understanding

• State the order of a given equation and justify it by naming the derivative that fixed it
• Decide whether degree is defined for a given equation, giving the reason in one sentence
• State the degree of a given equation when it exists
• Give both labels for each of ten mixed equations — the form of Exercise 9.1
• Choose the degree of an equation from four options, one of which is "not defined" — the form of Exercise 9.1 question 11
• Choose the order of an equation from four options — the form of Exercise 9.1 question 12
• Construct an equation of stated order and stated degree
• Construct an equation of stated order whose degree does not exist, and say why

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• The three order examples (§9.2.1, Part II pp. 301–302), numbered six, seven and eight by the chapter. The first sets a first derivative equal to an exponential; the second adds a second derivative to y; the third adds a third derivative to the square of the independent variable times the cube of a second derivative. The chapter reports orders one, two and three. Verified by inspection: correct in all three cases.
• Equation eight, on its own (Part II p. 302). This is the single most useful object in the topic and deserves a section to itself. Read on the printed page: the third derivative sits in a bracket carrying no exponent, while the second derivative sits in a bracket carrying a cube. Verified against the chapter's own two definitions: the order is three, because the third derivative is present; the degree is one, because the power on that derivative is one. The chapter itself lists this equation among those of degree one two paragraphs later, which confirms the reading. A student scanning for the tallest exponent answers three and three, and both halves of that answer are wrong for different reasons.
• The three degree test equations (§9.2.2, Part II p. 302), numbered nine, ten and eleven. The ninth mixes a third derivative, the square of a second derivative and a first derivative. The tenth squares a first derivative, adds the plain first derivative, and subtracts the square of the sine of y. The eleventh adds a first derivative to the sine of a first derivative. Verified: the ninth and tenth are polynomials in the derivatives they contain; the eleventh is not, because a derivative sits inside a sine. The tenth is the instructive one, and the chapter says so in a parenthesis — it is not a polynomial in y, and that does not matter, because the condition is about the derivatives.
• The degree verdicts (Part II p. 302). The chapter runs its own list: equations six, seven, eight and nine are each of degree one; equation ten is of degree two; equation eleven has no degree. Verified item by item: six has a first derivative to the first power; seven a second derivative to the first power; eight as argued above; nine a third derivative to the first power, the square sitting on a lower derivative exactly as in eight; ten a first derivative squared, which is also the highest derivative present, so the square does count; eleven fails the condition.
• The boxed Note (Part II p. 302). Both labels, when degree exists, are whole numbers greater than zero. Worth thirty seconds because it rules out the two answers students reach for when an equation resists — a fractional degree from a square root, and a degree of zero.
• Example 1 (Part II p. 303), three parts. Part one: a first derivative minus the cosine of the independent variable. Part two: a second derivative multiplied by x and y, plus x times the square of a first derivative, minus y times a first derivative. Part three: a third derivative, plus the square of y, plus the exponential of a first derivative. Verified: part one is order one, degree one; part two is order two, degree one — note that the square is on the first derivative, not the second, which is the same trap as equation eight in a new dress; part three is order three with no degree, the exponential playing exactly the role the sine played in equation eleven. The chapter reaches the same three verdicts on the same page.
• Exercise 9.1, questions 1 to 10 (Part II pp. 303–304). Verified, working added here on each: item one, a fourth derivative added to the sine of a third derivative — order four, degree not defined; item two, order one, degree one; item three, the fourth power of a first derivative in s added to three s times a second derivative — order two, degree one; item four, the square of a second derivative added to the cosine of a first derivative — order two, degree not defined; item five, order two, degree one; item six, squares, cubes and fourth powers of the third, second and first derivatives together with a fifth power of y — order three, degree two; item seven, order three, degree one; item eight, order one, degree one; item nine, order two, degree one; item ten, a second derivative, twice a first derivative, and the sine of y — order two, degree one, because the sine wraps y and not a derivative. Items four and ten are the pair to put side by side: they differ only in what sits inside the trigonometric function, and the verdict flips. The number of prime marks in item one was confirmed on the printed page, since at pack resolution a third prime is easy to lose.
• Exercise 9.1 questions 11 and 12 (Part II p. 304), both multiple choice. Item eleven asks for the degree of an equation carrying the cube of a second derivative, the square of a first derivative, the sine of a first derivative and a constant. Verified: the sine of a derivative destroys the condition, so the answer is the "not defined" option, which is the fourth. Item twelve asks for the order of an equation whose highest derivative is a second one. Verified: the answer is two, the first option. Together these two items are the whole topic in eight lines and make the best closing exercise.
• Miscellaneous Exercise question 1 (Part II pp. 333–334), three parts. Verified: part one, a second derivative plus five x times the square of a first derivative, minus six y, equal to a logarithm — order two, degree one; part two, the cube of a first derivative minus four times its square plus seven y — order one, degree three, and this is the only place in the chapter where the degree exceeds two; part three, a fourth derivative minus the sine of a third derivative — order four, degree not defined, which is item one of Exercise 9.1 with the sign changed.
• The Summary bullets (Part II p. 336). The Summary runs to eight bullets and three of them belong to this topic — the second, third and fourth. Counted on the printed page. Read them as the revision card they are, and note that the two degree bullets are printed in the right order: the condition first, the measurement second.

Figures to have open

• An enlarged, annotated setting of equation eight (Part II p. 302) for section 4, with the exponent on the second derivative and the absent exponent on the third marked in different colours. This is the load-bearing image of the topic. The equation is the chapter's; the annotation is added here.
• A six-row table for section 8. Build it with the repo's DataTable component. Content is the chapter's own list of verdicts on Part II p. 302; layout is added here.
• A two-panel comparison for section 11: Exercise 9.1 item four beside item ten, with the argument of the trigonometric function highlighted in each.
• A three-row card for section 1 showing one equation with two empty label slots.
• No textbook figure can be redrawn: this chapter prints none. All thirty-eight pages were opened as page images; the only artwork in the chapter is the portrait and QR code on Part II p. 300.

Where this sits in the book

• NCERT Class 12 Mathematics, Part II, Chapter 9 "Differential Equations", §9.2.1 Order of a differential equation, and the three examples numbered six to eight, pp. 301–302
• §9.2.2 Degree of a differential equation — three test equations numbered nine to eleven, then the definition and the list of verdicts, Part II p. 302
• The boxed Note on both labels being positive integers, Part II p. 302
• Example 1, three parts with solutions, Part II p. 303
• Exercise 9.1, questions 1 to 12, Part II pp. 303–304
• Miscellaneous Exercise question 1, Part II pp. 333–334
• Summary, second, third and fourth bullets, Part II p. 336