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

0:00Here are four equations, written side by side. A quadratic. A trigonometric equation. A relation between two letters. And a fourth one. Three of these you have been solving for years. The fourth starts a whole subject, and the difference between them is a single structural feature. The fourth one contains a derivative. The first three contain none. That is it. That is the whole definition, and everything difficult that follows is a consequence of that one difference.

0:36So it is worth being exact about what the feature actually is, because the obvious version of it is wrong. The condition is not that a derivative appears somewhere. It is that a derivative of the unknown appears. Those are different, and the difference matters immediately. Take an equation carrying the letter d as an ordinary symbol, multiplying something. There is a d on the page and there is no derivative in it at all.

1:09Take an equation carrying a genuine derivative, but of a symbol that is not the thing you are solving for. Still not a differential equation. Now take the same shape with the derivative taken of the unknown instead. That one is. Three equations, and only the third qualifies. The test is structural, and it looks in one specific place. That is a claim about structure, so it was checked by machine rather than by eye.

1:43Every equation in this video was written down as a tree: numbers, symbols, the unknown, derivatives, and the operations joining them. Deciding whether an equation is differential then means walking that tree and looking for a derivative taken of the unknown. Nobody types the answer in. Put in front of the four opening equations, the walk finds a derivative of the unknown in exactly one, and none in three. Put in front of nine equations that do carry one, it finds all nine. Put in front of the three that do not, it finds none.

2:26And one of those nine carries its derivative on the far side of the equals sign, while another carries its only one underneath a square, so the walk really is reading the whole tree. Before anything else, name the two roles, because misreading them is the most expensive mistake available here. One symbol stands for the unknown. That is the dependent variable, and in almost everything you will meet it is called y.

2:58The other is the symbol the unknown is differentiated against. That is the independent variable, and it is usually x. A student who reads the fourth equation and starts isolating x has swapped the two roles, and every step after that compounds it. You are not solving for x. You are solving for y, and y is not a number. Here is how much that reading matters, and how little. Take the third opening equation, the relation between two letters, and encode it twice: once with its second letter as a plain unknown number, and once as an unknown function of the first.

3:42The two are genuinely different trees, and only the second holds an unknown function at all. But the verdict does not move. Neither is a differential equation, because neither contains a derivative of anything. There is one more split to make, and it is usually made in a sentence and then dropped. Count the symbols the derivatives are taken against. If they are all taken against a single one, the equation is called ordinary.

4:16If derivatives are taken against two or more different symbols, it is called partial, and partial equations are not studied here at all. Ordinary does not mean easy. It is a statement about how many independent variables there are, and nothing else. Twelve equations were sorted this way, by counting symbols rather than by reading a label written beside them. Seven come out ordinary, two partial, and three are not differential equations at all.

4:50Exchange the two labels and only the three carrying no derivative still fit; the nine that do carry one all fail. So the sorting is reading the equations. From here on, the shorter word means the longer one. Every equation in the rest of this is ordinary. The order of a differential equation is the highest derivative of the unknown that occurs in it. Measured by walking each tree for its deepest derivative, all twelve equations come out at the order they were written to have.

5:27And raise every expected order by one and not a single equation fits, so that is a measurement and not a count of how many equations there are. You will see the same derivative written three ways. As a fraction, with d y over d x. With a prime mark. And, past the third, with a subscript carrying the order. The prime marks stop being readable after three of them, which is the entire reason for the subscript.

5:59And that subscript is not a term of a sequence. It is the order of a derivative. The collision with sequence notation is real, and it is worth ten seconds of your attention. Three notations, one object. A student who thinks a prime mark is a different operation cannot read the exercises at all. Now the consequence, which is the reason this topic exists. When you solve an ordinary equation, the answer is a number. You mark it on a number line as a point, and you are finished.

6:39When you solve a differential equation, the unknown is a function. So the answer is a function, and a function is not a point. You draw it. A curve in the plane, and it has a name: the solution curve. This is not a change of packaging. It is a change of what kind of object an answer is, and it is what makes everything later feel unfamiliar. Here is that contrast measured rather than described. Three number equations were handed to a routine that walks a line and counts the places the equation reaches nought.

7:18The first has two such places. The second has none. The third has exactly one. A fourth was added whose two places are at no step of the walk at all, so they can only be found by the sign changing between two steps. The routine finds both. Which raises the question you will be asked over and over: given a function, is it a solution of this equation? The procedure has two moves and needs no cleverness at all. Differentiate as many times as the equation asks for. Substitute everything in. See whether the two sides agree.

8:02That is verification. It is not the same skill as finding a solution, and the exercises that ask for one do not ask for the other. Here is the important part, and it is the part that gets skipped. When a derivative is written down in that first move, it is being asserted. So in this video none of them are. Every derivative used in a substitution was measured against the rule above it, by taking difference quotients from both sides of the point and requiring the band to close.

8:38Twelve differentiation steps across six candidates, none of them wrong and none unreadable. A candidate offered with the wrong first derivative fails both of its steps. One whose first derivative is right and whose second is not passes one and fails the other. And one that refuses to answer on half the stretch leaves both steps unreadable, because a slope cannot be measured where half the neighbourhood is not there.

9:11Now substitute, and watch what agreement actually has to mean. Six candidates were put into the same second order equation and read at forty-one places across a stretch. Two of them close at all forty-one. Three close at none. And one closes at exactly one place and nowhere else. That last one is the trap. At one particular value of x the two sides do agree, exactly. Everywhere else they do not.

9:43It is not a solution. A solution has to make the two sides agree identically, for every value where both are defined, and one place out of forty-one is not that. Put the same six into a different equation and the split comes out differently, so the split is a reading of the equation and not a property of the candidates. There is one more thing hiding here. Whether the two sides agree cannot be judged by asking whether the leftover is small.

10:17A decaying candidate makes every term in the equation small far out along the stretch, so the leftover is small too, and any equation at all starts to look satisfied. Judge the leftover on its own and one candidate moves from closing nowhere to closing somewhere. So the leftover is measured against the size of the terms that produced it, which is the difference between a real agreement and an arithmetic accident.

10:50Something else falls out of the answer being a function, and it surprises people. There is usually not one solution. There is a whole family of them. Twelve different multiples of one decaying exponential were substituted into that same equation, one at a time. All twelve close at every one of the forty-one places. And they are twelve genuinely different functions, not one written twelve ways: all sixty-six pairs among them are apart somewhere on the stretch.

11:25To keep that honest, one impostor was thrown in among them, a function that closes at exactly one place. It is not counted with the twelve, which is how you know the test tells closing everywhere apart from closing somewhere. Put the same twelve into a different equation and not one of them closes everywhere. The family belongs to its own equation and to no other. So the answer to a differential equation is not a point, and usually not even a single curve. It is a family of curves, and the next topic is about the two numbers that pick one out of the family.

12:09None of this is quite as new as it sounds, because you have been solving equations of exactly this shape already. Here is the oldest one. Somebody hands you a rate of change and asks for a function whose derivative is that rate. Write that down and look at it. There is a derivative of the unknown on one side. That is a differential equation, and integration is how you solve it.

12:38Five rates were handed over and, for each, a function was offered as one whose derivative is that rate. All five check out, measured by difference quotients rather than by quoting a table. Hold each answer against the next rate along instead and all five fail, so the check is a check. Which means every integral you have ever done was already an instance of the object we are naming here. You just were not calling it anything.

13:11And that seed carries the family with it, for a reason you already know. Add a constant to a function and its derivative does not change. The derivative cannot see the constant. Twelve different constants were added to one function and each result checked as an answer to the same rate. All twelve pass. And the twelve are twelve different functions: all sixty-six pairs among them are apart. So the constant is invisible to the derivative and visible to everything else.

13:47Hold the same twelve against a rate that has been given the constant, and only the one shifted by nothing fits. The check can tell the difference. That is where the family comes from, and it is why counting the arbitrary constants in an answer turns out to matter as much as the next topics say it does. A word about where these equations come from, because it is easy to assume every one has a story attached.

14:18Most of the equations you will be handed are pure algebra, with no situation behind them at all. But the subject is named as arising in physics, in chemistry, in biology, in anthropology, in geology and in economics. Anthropology and geology are the two nobody expects, and they are the reason the list is worth reading out rather than skimming. What those fields share is that they describe how something changes rather than what it is. A population growing, a quantity decaying, a temperature falling towards the air around it.

15:00You are handed the rate. You want the quantity. That is the shape of every one of them, and it is the shape of the seed we just wrote down. So, the whole of it, in four sentences. An equation is differential when a derivative of the unknown occurs in it, and not merely when a derivative or a letter d appears somewhere. The unknown is a function, so the answer is a curve rather than a point, and usually a family of curves rather than one.

15:34Checking a candidate is differentiation followed by substitution, and agreement means agreement everywhere, not at some convenient place. And you have been doing this since you first reversed a derivative. The only thing that is new is the name, and the machinery for the cases where reversing is not enough.