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

0:00Solve for x. You have been doing that since you were small, and the answer is a number, or a short list of numbers. Now here is the other job. Solve for y, where y is not a number but a function, and what you are given is a relationship between that function and its own rate of change. The answer is not a number. The answer is a function. And usually it is not one function either. It is a whole family of them.

0:32That last part is where the trouble starts, because a family needs describing, and describing it means writing down letters that stand for anything at all. How many such letters? That is the whole of this video, and the surprising thing is that the answer is not obvious, and is very often not given. First, the easy half. Given a candidate function, how do you check whether it belongs? You substitute. Put the function in place of the unknown, put its derivatives in place of the derivatives, and see whether the two sides agree.

1:13One warning, and it matters more than it looks. When you check whether what is left over is nothing, you have to judge it against the size of the things that made it. Take a function that decays. Far out to the right it is tiny, and so is everything built from it. Judge the leftover on its own, and a wrong equation looks satisfied. Judged against the terms, the same wrong equation fails at every one of those places, and the right equation still passes.

1:47Nothing is nothing relative to something. That rule is used everywhere in what follows. Here is an equation to work with. The second derivative of y, added to y itself, equals nought. And here is a candidate: a number times the sine of x, plus a second number times the cosine. Differentiate twice. The sine comes back with a minus sign, the cosine comes back with a minus sign, and adding y back cancels both.

2:23It closes. It closes for every choice of the two numbers, and it closes at every one of the twenty-one places it was checked at. Every derivative used in that check was measured, not quoted from a rule. The slope was read by watching the function actually move. So this is not one solution. It is a two-letter family of them, and every member is a solution. Now pin the letters. Let the first be two, and the second be nought.

2:57What is left is twice the sine of x. One function. No letters. Nothing left free to choose. It is still a solution. Substituting it still closes the equation, exactly as before. But something has been spent. Before, the family could be aimed. Now it cannot. And that is the distinction the two names are for: one of them still carries letters you may set, and the other has had them all set for you.

3:29The names. A solution still carrying unspecified constants is called the general solution. A solution whose constants have all been given values is called a particular solution. You will also meet an older pair of words for the same two things: the complete integral and the particular integral. Same distinction, older vocabulary. Read those two definitions again and notice what they say. One carries constants. The other does not. What neither of them says is how many.

4:08That silence is not harmless, because the count is exactly what gets asked. Take an equation whose deepest derivative is a fourth. How many unspecified letters must its general solution carry? Nought, two, three, or four? And the companion question. A particular solution of an equation whose deepest derivative is a third: how many arbitrary constants does it carry? Both questions have clean answers. But the rule that answers them, the rule that says the count of letters equals the order, tends to appear as a bare statement with no argument attached.

4:52A statement without an argument is a thing to memorise, and a thing you memorise is a thing you can forget in the wrong direction. So we are going to put the argument back. What follows next is a reconstruction. It is not a result you will be handed. We build it. Here is the idea, and it is a good one. Take two different members of a family, and look at them at one place. Compare their values there. Compare their slopes. Compare their second derivatives, and so on up.

5:29Suppose you find two members that agree on the value, and on the slope, and on everything below the j-th derivative, and still disagree at the j-th. Then no equation of order j can possibly hold for that family. An equation of order j takes the value and the derivatives below the top one and hands back the top one. Here it would owe two different answers to the same question.

5:59So climb. Keep going up while such a pair still exists. The moment there is no such pair, you have reached a derivative that the ones below it determine. That level is the lowest order any equation for the family can have, and nothing about the equation was used to find it. It came out of the family alone. Run that climb on a family carrying four letters. At the bottom, plenty of pairs disagree on the value alone. Climb one.

6:33Pairs still exist that agree on the value and part on the slope. Climb again. Pairs agree on both and part on the second derivative. Again: agree on three, part on the fourth. At every one of the four levels below the fourth, such a pair exists. And at the fourth derivative there is none. Once the value and the first three derivatives are fixed, the fourth is forced. The climb stops at four. Four letters, and the lowest order an equation can have is four.

7:09That is the first of the two questions answered, and answered without appealing to any rule: the count is four, measured off the family, and it matches the order of the equation, read off the equation separately. One honest caution before going further, because the climb has a soft spot. Take the family of straight lines through the origin: a number times x. Two letters? No, one. Read that family at x equals nought. Every member passes through the same point. Every pair agrees on the value there, and none of them parts, so the climb never even starts and reports no order at all.

7:56Step one unit along and read again. Now every pair parts immediately, and the climb reports one, which is correct. The place you stand can hide a difference. So when this argument is run, it has to be run somewhere the members are not conspiring to agree, and that is a real condition, not a technicality. The climb shows that a family with k letters cannot be pinned by fewer than k conditions. Now the other direction: can it be pinned by exactly k?

8:33Demand a starting value. Demand a starting slope. For a family with two letters, demand both. Across six families, and across every combination of demanded values tried, the answer came back the same way. A hundred and thirty-eight demands, every one of them met. And met properly. The member was proposed by arithmetic, and then actually evaluated to check that it really did start where it was told to. Now ask for one value more than there are letters. Four hundred and fourteen such demands. A hundred and thirty-two were met, and two hundred and eighty-two were not.

9:16The extra condition is already spoken for. The family has no coin left to pay it with. Put those two halves together and you have the rule, earned rather than asserted. Six families were built, each carrying a different arrangement of letters. For each one, three separate readings were taken. The order of its equation, read straight off the equation. The count of unspecified letters it carries. And the order the family forces, measured by the climb.

9:52All six agreed on all three. And they are not six copies of one number: four different orders appear among them. That is what the rule means. A general solution of an equation of order n carries n arbitrary constants, and each of those three ways of counting gives the same n. Now the case that stops this becoming a slogan, because it is the one that comes apart. Here is a single function: e to the minus three x. It carries no letter at all.

10:28Substitute it into a particular second-order equation and it closes, everywhere, cleanly. It is a solution. But run the climb on it and there is nothing to climb. One function, no pairs, no disagreement anywhere. The order it forces is nought. An equation of order two, a genuine solution, and no arbitrary constant. Verifying a solution and having a general one are two different things. And here is the sharp version. That lone function is one member of a two-letter family that closes the very same equation — the two agree at all twenty-one places — while a different member of that family agrees with it at none.

11:18So the count of letters belongs to the solution you are holding. It never belongs to the equation. Which also answers the second question from earlier: a particular solution carries no arbitrary constants at all, whatever the order. Nought, for an equation of order three, and nought for any other. One more thing worth seeing, because it shows that a general solution is a set of functions, not a particular way of writing them.

11:52You will meet the family of a sine and a cosine, each with its own coefficient. You will also meet a family written as a single sine with a size in front and a shift inside it. These are the same set. Not similar. The same. Every one of the twenty-five shifted sines was matched to a pair of coefficients that agrees with it at all twenty-one places. And every one of the twenty-four coefficient pairs was matched back to a size and a shift.

12:27The matching backwards was done by halving — searching for the shift numerically — rather than by quoting the expansion of a sine of a sum, because that expansion is the very thing being tested. And it is a real match. Pair each member instead with the coefficients belonging to its neighbour, and not one of the twenty-five agrees. Same set, then. But the two ways of writing it are not equally convenient, and that is worth knowing before an exam.

13:02Take nine demands for a starting value and a starting slope. The sine-and-cosine writing supplies a member for all nine. The shifted-sine writing supplies a member for six. The three it misses are all the same kind: the ones that ask the curve to pass through nought, where the shift cannot be pinned by the search. The set of solutions is unchanged. What changes is how easily you can reach into it and pull out the member you were asked for.

13:35So when a question hands you a general solution in one form and a condition in the other, changing form first is sometimes the whole of the work. Now the problems that ask you to do this, run down quickly. Five of them hand you a formula for y. One of those carries no letter at all and stands against an equation of order two — that is the warning case again, sitting quietly in a list. The other four carry one letter each, against equations of order one.

14:12Three more hand you not a formula but a relationship between x and y, tangled together. For those, the trick is that each one can be untangled the other way round: x written in terms of y. Then measure how x moves as y moves, turn the reading over, and you have the slope. Seventeen of the eighteen readings taken that way close their equation. Not one fails. One is refused, and it is refused at exactly the place that problem excludes on its own terms.

14:49And to be sure that is agreement with something: move the constant in each of those equations by a half, and not one of the seventeen closes. So, the whole of it. A solution is a function that closes the equation when you substitute it, and closing it is checked against the size of the terms, never on its own. A general solution still carries letters you may set. A particular one has had them all set.

15:21The count of those letters equals the order, and the reason is not a rule to be memorised. Two members of the family agreeing all the way up to some derivative and parting there make an equation of that order impossible; the level where the parting stops is the order the family forces. A general solution of an equation of order four carries four constants. A particular solution carries none, whatever the order.

15:53But closing the equation is not the same as being general — a lone function with no letters can do the first and not the second. And the general solution is a set of functions. Two very different-looking formulas can name the same set, and choosing between them is a matter of convenience, never of correctness.