General against particular: why the arbitrary constants are counted by the order

What to assume they know

• Order of a differential equation, from the previous topic
• Differentiating a function that carries unspecified constants, treating them as constants
• Substituting an expression into an equation and simplifying both sides
• The derivatives of sine, cosine and the exponential
• Implicit differentiation, for the exercise items given as relations rather than formulas
• Reading a family of curves as one equation with a parameter left free

What they should be able to do

• Explain why the answer to a differential equation is a function and not a number, and say what substituting it is supposed to achieve
• Verify a proposed solution by differentiating the required number of times and substituting
• Handle a proposed solution given implicitly, differentiating the relation rather than a formula
• State both definitions the chapter prints, and identify which is which from the presence or absence of unspecified constants
• Recognise the bracketed older name the chapter gives the general solution
• Count the arbitrary constants in a proposed general solution and compare the count with the order
• Argue for the rule that the two numbers match, from what one differentiation can eliminate
• Produce a particular solution from a general one by fixing the constants to named values
• Say how many arbitrary constants a particular solution carries, whatever the order

Where it usually goes wrong

• "The general solution is a single function." It is a set of functions, one for each choice of the constants. The chapter's two different-looking families, printed a page apart on Part II pp. 304 and 305, are the same set, which is the strongest available argument that the formula is not the object.
• "A particular solution is a special kind of function." It is any member of the family, once the constants have been chosen. Nothing distinguishes it intrinsically; the chapter's own particular solution on Part II p. 305 is just the family with two named values put in.
• "Any solution with no letters left in it is a particular solution of that equation only." True as far as it goes, but students then conclude that a solution with no constants cannot be checked. Example 2 is exactly such a function, and it is checked in four lines.
• "A second-order equation needs two conditions because it has two constants — so the count is a definition." It is not a definition; it is a fact that has a reason, and the reason is that each differentiation is one opportunity to eliminate one constant. Because this chapter nowhere prints the procedure that makes that visible, an explanation that states the rule flatly is passing on an unexplained slogan.
• "More constants means a more general answer, so I should add one." Adding a constant that the equation can absorb produces a family that is not a solution set, and adding one it cannot changes nothing. The count is fixed by the equation, not chosen by the solver.
• "Verifying is trivial, so I can skip it." Three of the ten items in Exercise 9.2 are given implicitly, and implicit differentiation is where the marks are lost. Item nine needs the relation differentiated before anything can be substituted, and item ten, though explicit, is far easier squared first than differentiated as printed — so four of the ten reward not taking the direct route.
• "If both sides agree at the value I tried, the function is a solution." The agreement has to be identical in x. Every worked verification in this section ends with an expression that cancels symbolically, never with a number substituted in.
• "The bracketed word beside the general solution is a different concept." It is an older name for the same thing, and it returns once more inside the homogeneous method on Part II p. 314. Students who meet it there for the first time assume a new object has appeared.

Questions to check understanding

• Verify that a given explicit function satisfies a given equation
• Verify that a given implicit relation satisfies a given equation — the form of Exercise 9.2 questions 7 to 9
• Classify a given solution as general or particular, and justify from the definition
• Given an equation of stated order, say how many unspecified letters its general solution carries — the form of Exercise 9.2 question 11
• Say how many unspecified letters a particular solution carries — the form of Exercise 9.2 question 12
• Produce a particular solution from a stated general one by fixing the constants
• Given a family and a point, decide which member passes through the point
• Explain in two sentences why the count of constants matches the order

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 two number equations that open §9.3 (Part II p. 304). One quadratic and one built from a squared sine and a cosine. The chapter uses them only to say what an answer looks like when the unknown is a number: a value that makes the two sides agree once it is put in.
• The chapter's running equation (Part II p. 304, numbered three). A second derivative added to y, set to zero. It is worth marking that the same equation reappears as Example 3 on the very next page with a differently written solution — see below. Building the topic around one equation seen twice is the cleanest structure available here and it is the chapter's own doing.
• The two-parameter family (Part II p. 304, numbered four). A constant times the sine of the sum of x and a second constant, both constants free over the reals. The chapter asserts, without showing the working, that substituting this and its derivative closes the equation. Verified: differentiating twice multiplies the sine by minus one, so the second derivative is the negative of the function and the sum is zero for every x, whatever the two constants are.
• The particularisation (Part II pp. 304–305, numbered five). The chapter sets the first constant to two and the second to a quarter turn, and observes that the resulting function still satisfies the equation. That is the whole mechanism of the topic: the family solves the equation, so each member does.
• The pair of definitions (Part II p. 305). Printed as two consecutive sentences. The first says a solution carrying arbitrary constants is the general one, and gives an older name in brackets. The second says a solution carrying none — obtained by fixing the constants — is a particular one. Verified on the printed page: the first sentence contains no clause about how many constants there must be. This is the finding the topic is built around, and it is recorded again in Notes.
• Example 2 (Part II p. 305). A decaying exponential against a second-order equation. Verified: the first derivative is minus three times the function, the second is nine times it, and nine minus three minus six is zero, so the left side closes identically. Note: this function carries no arbitrary constant at all, so by the chapter's own definition it is a particular solution of a second-order equation — a good, quiet demonstration that verification and generality are independent questions.
• Example 3 (Part II pp. 305–306). A constant times the cosine plus a constant times the sine, against the equation from Part II p. 304. Verified: differentiating twice returns the negative of the function, so the sum is zero for all x and both constants survive untouched. And verified further, as the point of section 3: this two-parameter family and the one on Part II p. 304 are the same set of functions written two ways, since expanding the sine of a sum turns one constant times the sine of x plus a shift into two constants multiplying the sine and the cosine separately. The chapter prints both and never remarks on it.
• Exercise 9.2, questions 1 to 10 (Part II p. 306). Ten verification items. Counted on the printed page: three of the ten are given implicitly — items seven, eight and nine — and the other seven are explicit formulas for y, including item ten, which follows the implicit run and returns to an explicit square root. The instruction line says either kind may appear and does not say how many of each. Verified, working added here: item one, an exponential plus one against a second-order equation — both derivatives equal the exponential, so their difference is zero. Item two, a quadratic with one constant, against a first-order equation. Item three, a cosine plus a constant. Item five, a constant times x against x times the first derivative equalling y — the first derivative is that constant, so multiplying by x reproduces the function. Item nine, a relation setting the sum of x and y equal to an inverse tangent of y — differentiating gives one plus the first derivative equal to the first derivative over one plus y squared, and clearing the denominator leaves exactly the printed equation. Item ten, the upper half of a circle of radius a against x plus y times the first derivative — squaring first and differentiating gives twice y times the first derivative equal to minus twice x, which is the printed equation. Count the constants as you go: items two, three, five and seven carry one each and their equations are all of order one; item one carries none and its equation is of order two. The exercise is quietly full of evidence for the rule the section never states.
• Exercise 9.2 questions 11 and 12 (Part II p. 306), both multiple choice. Item eleven takes an equation of order four and asks how many unspecified letters its general solution must carry; the options run zero, two, three, four. Verified: the answer is four, the last option. Item twelve asks how many sit in a particular solution of a third-order equation; the options run three, two, one, zero. Verified: the answer is zero, again the last option, and it follows straight from the printed definition rather than from any count. These two items are the only place in the body of the chapter where the count is examined, and neither the section above them nor any example on the way establishes it.
• The Summary bullet (Part II p. 336). One bullet does three things: it says a function satisfying the equation is its solution; it says a solution carrying one arbitrary constant for each unit of order is the general one; and it says a solution carrying none is particular. Verified on the printed page: the count-of-constants clause is present here and, as recorded above, absent from §9.3. A student revising only from the Summary meets a stronger rule than the chapter taught, and a student revising only from §9.3 cannot answer question eleven of the exercise.
• The phrase to steal for section 10 (Example 7 solution, Part II p. 309). The chapter says in passing that its general solution stands for a whole family of solution curves, and that the point given picks out one member of that family. The same idea returns near the end of the linear method (Part II p. 328), where a general solution is again described as an equation of a family of curves with one member wanted. Checked across the extracted text of all thirty-eight pages: the word for a family occurs five times in the chapter and in four places — those two, the statement of Example 13 (Part II p. 319, a family named by the slope of its tangents) and its solution on Part II p. 320. A fifth locution does the same work without the word, in Exercise 9.3 question 16 (Part II p. 311), which asks for the solution curve through a stated point. The chapter draws this picture in words in four places and on paper in none. Part II pp. 309 and 328 are the two worth quoting, because they are the only ones that say out loud that the general solution is the family.

Figures to have open

• A side-by-side frame for section 1: a number line carrying one marked point, against a pair of axes carrying one curve. Reuse the frame built for m01-t01 section 7 so the two topics visibly share a spine.
• A drawn family for section 10: at least five members of the chapter's own two-parameter family on one set of axes, with one member thickened and a point marked on it. The chapter draws nothing of the kind — all thirty-eight pages were opened as images and this chapter prints no figure at all — so this is an added construction and is the single most valuable picture in the module.
• An elimination diagram for section 7: a family carrying two constants at the top, an arrow labelled with one differentiation, one constant crossed out, and the same again below. Not in the book; nothing like it survives in this edition.
• A table for section 11 built with the repo's DataTable component, listing Exercise 9.2 items against the order of each equation and the number of constants in each given function.
• No textbook figure can be redrawn.

Where this sits in the book

• NCERT Class 12 Mathematics, Part II, Chapter 9 "Differential Equations", §9.3 — the two number equations, the running second-order equation, the naming of the solution curve, pp. 304–305
• The two-parameter family and its particularisation, Part II pp. 304–305
• The two printed definitions and the bracketed older name, Part II p. 305
• Examples 2 and 3 with their verifications, Part II pp. 305–306
• Exercise 9.2, questions 1 to 12, Part II p. 306
• The family-of-curves remarks inside Examples 7 and 18, Part II pp. 309 and 328
• Summary, the solution bullet, Part II p. 336