What makes an equation differential, and why the answer is a curve not a number

What to assume they know

• Differentiation of standard functions, from Class XI and from Chapter 5 of Part II's companion volume
• Indefinite integration as the reverse of differentiation, from Chapter 7
• The meaning of dependent and independent variable in a function
• Function notation, and evaluating a function at a point
• Sketching a curve from its equation, and reading a point off it
• Solving an ordinary algebraic or trigonometric equation for a number

What they should be able to do

• Decide, for a given equation, whether a derivative of the unknown appears in it, and classify the equation accordingly
• State in your own words the condition an equation must meet to be called differential
• Identify which symbol is playing the dependent role and which the independent role in a given equation
• Distinguish an equation carrying derivatives with respect to one variable from one carrying derivatives with respect to several, and name the two kinds
• Explain the chapter's decision to drop a qualifier from the phrase it uses throughout
• Read and write the prime notation for the first three derivatives, and the subscript notation the chapter reserves for higher ones
• Explain why an unknown that is a function forces the answer to be a curve rather than a number
• Test a proposed function by substituting it, and say what agreement of the two sides establishes
• Name several fields in which such equations arise, from the chapter's own list
• Place the subject historically using the portrait, the epigraph and the dates the chapter supplies

Where it usually goes wrong

• "A differential equation is any equation with a d in it." What is required is a derivative of the unknown. An equation can carry the letter d, or a differential written loosely, and still be an ordinary relation between numbers. The chapter's own test is structural: does a derivative of the dependent variable occur.
• "I am solving for x." The unknown is the function, and in this chapter's notation that means y. A student who reads the fourth opening equation and starts isolating x has misread which symbol is the unknown, and every later step will compound it.
• "So the answer is a number after all, once I finish integrating." The answer is a function, and typically a whole family of them. Integrating is how the answer is found; what is found is still a rule, not a value.
• "Ordinary means easy and partial means hard." Ordinary means the derivatives are all with respect to one variable. It is a statement about how many independent variables there are, not about difficulty, and the chapter says so in one sentence before dropping the qualifier for good.
• "The prime notation and the fraction notation mean different things." They name the same derivative. The chapter's Note exists because the same object gets written three ways within two pages, and a student who thinks a prime mark is a different operation cannot read Exercise 9.1 at all.
• "A subscript on y means a term of a sequence." In this chapter it means the order of the derivative, and the Note says so explicitly. This is a real collision with the sequence notation from Class XI and is worth ten seconds.
• "Verifying a solution and finding one are the same skill." Verification is differentiation followed by substitution and needs no cleverness; finding is the subject of the second half of the chapter. Exercise 9.2 asks only for verification, and a student who does not notice will try to solve every item.
• "If the two sides agree at one value of x, the function is a solution." It must agree identically, for every x where both sides are defined. Example 2 reaches zero as an algebraic identity in x, not by evaluating at a point.
• "Every differential equation comes from a physical situation." Most of the chapter's equations are handed over as pure algebra. The list of fields on the opening page says where they arise, not that each printed one has a story attached.

Questions to check understanding

• Given a mixed list of equations, pick out the ones carrying a derivative of the unknown
• Name the dependent and independent variable in a given equation
• Say whether a given equation is of the ordinary kind, and justify the answer by counting independent variables
• Rewrite a derivative from fraction notation into prime notation and back
• Rewrite a fifth derivative using the subscript convention
• Verify by substitution that a given function solves a given equation — the form of Exercise 9.2 questions 1 to 10
• Explain in one sentence why the answer to a differential equation is drawn rather than marked on a number line
• State one field, from the chapter's list, in which such equations arise

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 four numbered equations of §9.2 (Part II pp. 300–301). Read off the printed pages, the chapter's first three are a quadratic in one unknown, a trigonometric equation in one unknown, and a linear relation between two unknowns; the fourth multiplies the first derivative by the independent variable and adds the dependent one, and sets the result to zero. Verified by inspection: the first three contain no derivative symbol at all; the fourth contains exactly one. That single difference is the entire definition.
• The seed equation of §9.1 (Part II p. 300). The chapter's opening move is to restate the integration problem as a question about an unknown function: a function g is handed over, and a function whose derivative is g is wanted. It then observes that what has just been written down is already an instance of the object about to be named. This is the cleanest possible entry point, because the student has been solving equations of exactly this shape since Chapter 7 without calling them anything.
• The general condition (§9.2, Part II p. 301). The chapter states it twice — once about the specific fourth equation, once in general — and both statements hinge on the same clause: a derivative of the dependent variable, taken with respect to the independent one, must occur. Note that the chapter writes the plurals in brackets, which reads awkwardly aloud; say "one or more derivatives" and "one or more independent variables".
• The ordinary-against-partial split (§9.2, Part II p. 301). One displayed example is offered for the ordinary kind: twice the second derivative added to the cube of the first, set to zero. The chapter then says partial equations exist, declines to study them, and announces that from that point on the shorter phrase will mean the longer one. Verified by reading every page of the chapter as a page image: no partial derivative symbol appears anywhere in Part II pp. 300–337 after that sentence. The promise is kept.
• The Note on notation (Part II p. 301). Two items. The first fixes prime marks for the first, second and third derivatives. The second explains that stacking further prime marks becomes unreadable, so a subscript carrying the order is used instead from the fourth onwards. Both notations are then used live: Exercise 9.1 asks about a fourth derivative written the long way in the same line as a third derivative written with primes (Part II p. 303).
• What a solution has to do (§9.3, Part II p. 304). The chapter draws the contrast the explanation needs: for its two number equations the answer is a number which, substituted for the unknown, makes both sides agree; for a differential equation the answer is a function which, substituted for the dependent variable, does the same job. Then it names the graph of that function the solution curve, giving integral curve as a second name in brackets.
• Example 2 (Part II p. 305). The function is a decaying exponential and the equation is second order. The chapter differentiates twice, substitutes all three quantities, and the left side collapses to zero. Verified: with y equal to e to the power minus three x, the first derivative is minus three times that and the second is nine times it; nine minus three minus six is zero, so the substitution does close. Use this as the explanation's demonstration of section 8 — it is the shortest complete verification in the chapter.
• The list of fields (§9.1, Part II p. 300). The chapter names Physics, Chemistry, Biology, Anthropology, Geology and Economics, in that order, and then says the subject has therefore become important across modern science. Anthropology and Geology are the two a student will not expect; keep them, as they are the reason the list is worth reading aloud at all.
• The chapter frontispiece (Part II p. 300). A QR code lettered with the Part II catalogue number and the chapter number, an epigraph attributed to D. Hilbert about method without a problem to aim at, and a portrait captioned with Poincaré's name and the years 1854 to 1912. Verified on the printed page: the QR lettering reads twelve-thousand-eighty followed by the chapter tag, which is Part II's catalogue number and independent confirmation of the volume. Caption these on a card; do not reproduce the page.
• The Historical Note, first paragraph only (Part II p. 336). The subject is given a birthday in November 1675, attached to the occasion on which Leibniz first wrote down an integral of y with respect to y and thereby introduced both the integral sign and the differential symbol. Two sentences of this make a strong end card for this topic. Note the printed spelling slips recorded below before any name is shown.

Figures to have open

• A build of the four opening equations for section 1, greying the first three as the fourth arrives. The equations are the chapter's own (Part II pp. 300–301); the treatment is added here. This chapter prints no figure of any kind, so every drawing listed here is an added construction.
• A single-frame contrast for section 7: a number line with one point marked beside a set of axes with one curve drawn. This is the load-bearing image of the topic and should be reused whenever the point recurs.
• A three-row notation card for section 6, the same derivative in fraction form, prime form and subscript form.
• A caption card for the frontispiece (Part II p. 300) naming the portrait and its dates and quoting the epigraph's attribution. Ignore the QR code. Spell Poincaré with its accent; the page does not.
• No redraw of any textbook figure is possible or required. Pages 300 to 337 were all opened as page images and the only artwork in the chapter is the portrait on the opening page and the QR code beside it.

Where this sits in the book

• NCERT Class 12 Mathematics, Part II, Chapter 9 "Differential Equations", §9.1 Introduction, p. 300
• §9.2 Basic Concepts, the four numbered equations and the general condition, Part II pp. 300–301
• The ordinary-against-partial paragraph and the boxed Note on notation, Part II p. 301
• §9.3 opening paragraphs, the contrast with number equations and the naming of the solution curve, Part II p. 304
• Example 2, Part II p. 305; Exercise 9.2 questions 1 to 10, Part II p. 306
• Summary, first and fifth bullets, Part II p. 336; Historical Note, first paragraph, Part II p. 336