Differential Equations

MIT 18.01 Single Variable Calculus, lecture 15, explained in plain English

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Part 1 · 0:00–10:00

This part shows how to solve a simple type of differential equation, one where the rate of change depends on x and y, by separating the variables and integrating.

Key ideas

A differential equation is an equation that links a function to its rate of change. The simplest kind, dy/dx = f(x), is solved by taking the antiderivative of f(x). 2:12
When the rate of change depends on both x and y, you cannot integrate straight away. Instead, you rearrange so that all the y terms sit on one side with dy and all the x terms sit on the other side with dx. 5:59
Once the variables are separated, you integrate both sides. Then you use exponentials to get y by itself, which gives a family of answers with a constant in them. 7:38
Any constant multiple of e^(-x^2/2) solves the equation. You can check this by taking the derivative and using the chain rule. 9:09

Words to know

differential equation: An equation that tells you how fast something changes, so you have to work backward to find the original function.
antiderivative: The reverse of a derivative. If you know how a quantity is changing, the antiderivative tells you the quantity itself, up to a constant.
separate (the variables): Rearranging an equation so that every y term is on one side and every x term is on the other, so each side can be integrated on its own.
chain rule: A rule for taking the derivative of a function built inside another function, such as e raised to a power that itself depends on x.

Example

Imagine a tank that drains faster the more water it holds. Knowing how fast the water level is falling at each moment is like knowing the differential equation, and working out the water level over time is like finding the antiderivative. In this lesson, the rate of change depends on both x and y, so you first sort the terms into separate sides before you can work backward to the answer.

3 checked practice questions for this part are in the app.

Part 2 · 10:00–20:00

This part explains when the separation of variables method works, the steps for using it, and two common traps: a solution that gets lost along the way, and the constant that shows up in the answer.

Key ideas

Separation of variables works when a differential equation has the form f(x) times g(y). Each side can then be handled on its own. 10:53
The key step is moving all the y terms to one side and all the x terms to the other, by dividing by g(y) and multiplying by dx. Then you take the antiderivative of each side. 11:42
The answer often comes out as an implicit equation, which defines y without giving a direct formula. You can solve for y to get an explicit formula, but that step can be messy, so sometimes you leave it implicit. 13:10
The solution y = 0 can get lost, because the method divides by y. Dividing by zero is not allowed, so you have to check for that case separately. 16:40
The constant starts out added, but after you apply nonlinear steps like exponentiation, it can become multiplied. Also, two separate constants can always be combined into one. 19:05

Words to know

implicit equation: An equation that links x and y without stating y by itself, like a rule that y must satisfy.
antiderivative: The reverse of a derivative. It is a function whose rate of change gives back the original function.
inverse: The operation that undoes another one, such as taking a square root to undo squaring.
arbitrary constant: An unknown fixed number added to an answer, which can be any value, because many answers differ only by it.

Example

Imagine sorting a box of mixed items into two piles, one for the x items and one for the y items, so each pile can be counted separately. This is like how the method separates x and y so each side can be integrated on its own. The lesson does not use this example; it is only meant to make the idea easier to picture.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part shows how to solve a differential equation by separating the variables, how to handle the many solutions it can produce, and how to use the method on a curve-shape geometry problem.

Key ideas

Separation of variables means moving all the y pieces to one side and all the x pieces to the other, so you get dy = f(x) dx. Then you integrate both sides to find the solution. 20:21
Solving a differential equation often gives a whole family of answers, not just one. The answer is written as a general solution that includes a parameter, which is a constant that can take different values. 21:53
If you are also told the value the function starts at, such as 3, you can plug it in to find the parameter and pick out one exact function. 22:44
A simpler version of the method can miss some solutions, such as the case where the parameter is 0. You have to think about these missing cases separately so you do not lose any. 23:53
Some geometry problems about curves can be written as differential equations. For example, a rule that the tangent line is twice as steep as the ray from the origin becomes dy/dx = 2(y/x), which you can then solve by separating variables. 27:19

Words to know

Separation of variables: A method for solving equations by moving all the y parts to one side and all the x parts to the other, then integrating each side.
General solution: The full set of all possible answers to a differential equation, written with a constant that can be any value in the family.
Parameter: A constant in a general solution that stands for a whole family of answers. It can be fixed by extra information about the function.
Antiderivative: The result of reversing a derivative, which is the function you started from before it was differentiated. Integrating finds an antiderivative.

Example

Think of a bathtub filling with water at a rate you know, but you do not know how much water was in it at the start. The rate alone gives you a whole family of possible water levels, one for each starting amount. If someone tells you the water was at 3 inches at the beginning, you can pick out the one exact level that matches.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–40:00

This part shows how to solve differential equations by separating variables, and how the answers can gain or lose cases depending on the problem.

Key ideas

The first example's rule says the tangent line's slope is twice the slope of the line from the origin to the point. Solving it gives a whole family of parabolas, y = ax^2, where a is any constant. 30:20
Using absolute values covers both the upward and downward curves. The answer becomes y = Ax^2 or y = -Ax^2, depending on the sign. 31:45
Dividing by y in the method hides the case y = 0, so the method misses that answer. The a = 0 case is also not picked up by this method. 33:24
The equation is undefined at x = 0, so knowing the value and the rate of change there does not fix the function. It is better to think of the answers as rays from the origin. 34:05
For curves perpendicular to the parabolas, the slope is the negative reciprocal, giving dy/dx = -x/(2y). Solving gives x^2/2 + y^2 = c, but only positive values of c work, and those give ellipses. 36:44

Words to know

tangent line: A straight line that just touches a curve at one point and has the same slope as the curve there.
antiderivative: The result of working backward from a rate of change to find the original function.
negative reciprocal: The number you get by flipping a fraction upside down and changing its sign, such as the negative reciprocal of 2 being -1/2.
parameter: A constant in a formula that can take different values to give a family of related curves.

Example

Think of a set of hills where every hilltop lies on a line from a fixed point. Each hill is one curve in a family, and the rule tells you how steep each one must be at every point. The constant in the answer is like picking which hill you are standing on.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–45:23

This part explains how a differential equation describes a curve's shape, why an ellipse is not a function, and the main idea of the unit: information about a derivative tells you about the function itself.

Key ideas

The equation being studied produces a curve shaped like an ellipse, which is an egg-like oval that is a little longer than it is tall. The lecturer says the curves cross each other at right angles. 40:00
The implicit solution gives the shape directly. Solving it for y gives the explicit solution, which is a formula for y in terms of x. 41:06
An ellipse is not a function, because one x value can have two y values. Only the top half or the bottom half gives a solution that works as a function. 41:32
At y = 0 the solution reaches a vertical slope and stops. This fits the equation, because y appears in the denominator, so the slope becomes infinite there. 42:35
The main theme of the unit is that information about a derivative, and sometimes the second derivative, tells you about the function f itself. 44:51

Words to know

ellipse: A stretched-out oval shape, like an egg that is a little longer than it is high.
implicit solution: A solution written as an equation that links x and y together, without y being alone on one side.
explicit solution: A solution where y is written alone on one side, as a formula in terms of x.
function: A rule where each input x gives exactly one output y, so no x value gives two different y values.

Example

Think of a hill with a path around it. The path's shape tells you where you are going, and the steepness at each point is the derivative. Knowing how steep the hill is at every spot helps you draw the whole path, the same way the derivative helps describe the function itself. This is only a comparison; the lecture does not use this example.

3 checked practice questions for this part are in the app.

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Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

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