Lecture notes · Differential Equations · Lecture 1

The Geometrical View of y'= f(x,y)

MIT 18.03 Differential Equations, lecture 1, explained in plain English

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Part 1 · 0:00–10:00

This part introduces first-order differential equations and a way to understand them by drawing pictures instead of solving them with formulas.

Key ideas

Some first-order equations look simple but cannot be solved with ordinary formulas. The lecturer says these are the rule, not the exception, so he will show what to do with them. 3:25
A direction field is made by placing tiny line segments on the plane. Each segment's slope is set by the equation at that point, which is f(x,y). 7:04
An integral curve is a curve that, at every point, follows the direction of the field. It touches the tiny line segments as it passes through them. 8:37
Writing the equation and drawing the direction field are two views of the same thing. Finding a solution is the same as drawing an integral curve. 9:18
The lecturer says that y1 of x is a solution if and only if its graph is an integral curve. This links the two views together. 9:44

Words to know

direction field: A picture made of many tiny slanted line segments. Each one shows which way a solution would be heading at that spot.
integral curve: A curve that follows the tiny slanted segments all the way along. At each point, it goes in the same direction as the segment there.
line element: One tiny line segment drawn at a single point on the plane. Its only job is to show a slope.
elementary functions: Common functions you can write down with a formula, such as simple algebra and standard functions. The lecturer says some equations have no solution written this way.

Example

Think of a river with arrows painted on the water at many spots, each showing which way the current pushes. A leaf dropped in the river and carried along the current traces a path that always matches the arrows. That path is like an integral curve, and the arrows are like the direction field.

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Part 2 · 10:00–20:00

This part explains why a solution curve matches the direction field, and how a person can draw a direction field by first picking slopes (isoclines) instead of points.

Key ideas

A solution is a curve whose slope at every point equals the value the differential equation gives there. Checking this means plugging the function into the equation and seeing that both sides match. 10:42
A computer draws a direction field by picking evenly spaced points and computing the slope at each one. This works, but a human finds it tedious because every new point needs a new calculation. 14:23
A human usually starts by picking a slope C and finding all the points where the slope equals C. These points form a curve called an isocline. 16:18
Once you have an isocline, you draw short line segments with slope C along it. Doing this for several slopes fills in the picture, and you can add more segments if you need a clearer picture. 17:56
Isoclines are drawn as dashed lines, because they are not solutions. Solution curves are drawn as solid lines. 17:56

Words to know

integral curve: A curve that follows the direction field everywhere. At each point, its slope matches the arrow or line segment drawn there.
direction field: A picture made of tiny line segments, one at many points, each showing the slope the equation requires at that point.
isocline: A curve made of all the points where the slope is the same value C. Think of it as a 'same slope' path across the picture.
level curve: A curve where a function has the same value everywhere along it, like a contour line on a map that shows equal height.

Example

Think of a hiking trail on a hill where every spot has a sign showing how steep the ground is. An isocline is like tracing all the spots with the same steepness, such as every place that is exactly a 10 percent slope. Once you know those spots, you can sketch how the land looks without measuring every single point.

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Part 3 · 20:00–30:00

This part works through two examples of drawing a direction field by first finding the isoclines, then using them to sketch the solution curves.

Key ideas

An isocline is a curve where every point has the same slope. You find it by setting the equation for y prime equal to a constant C. 20:45
For y prime equals minus x over y, the isoclines are straight lines through the origin, such as y equals minus x when C is one. 21:26
The small line segments, called line elements, are always perpendicular to the isoclines. That is because their slopes are negative reciprocals of each other, so you can often skip extra calculation. 23:19
For this same example, the solution curves, called integral curves, are circles centered at the origin. A solution may only cover a limited part of the x-axis, and you often cannot tell that just by looking at the equation. 27:25
The equation y prime equals one plus x minus y cannot be solved by separating variables, so the lecturer can only get a geometric picture of its solutions, not an exact formula. 29:41

Words to know

isocline: A line or curve where every point has the same slope. Think of it as a path where the steepness never changes.
line element: A tiny line segment drawn on a graph to show the slope of a solution at one point.
integral curve: A curve that is a solution to the equation. It follows the slopes that the equation gives at every point.
separation of variables: A method for solving some equations by moving all the y terms to one side and all the x terms to the other, then integrating each side.

Example

Imagine a hiking map with contour lines that join points at the same height. Each isocline works like one of those lines, joining points that all have the same slope. Walking along one contour, you never climb or drop more steeply, just as a solution curve crossing an isocline has the same steepness there.

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Part 4 · 30:00–40:00

This part shows how the lecturer uses isoclines and line elements to sketch solution curves, and why those curves cannot cross, touch, or escape a certain strip.

Key ideas

Isoclines are lines where every line element has the same slope. For C equals zero, the isocline is y equals x plus 1, and other values of C give parallel lines. 30:13
A corridor between two isoclines acts like a trap. Once a solution enters, it cannot escape, because every slope inside points the same way it must keep going. 34:36
Two integral curves cannot cross at an angle. At any point the direction field gives only one slope, so two curves cannot both pass through it with different slopes. 36:35
Integral curves that get trapped move closer and closer to a line. Here the line y equals x is itself an integral curve, and plugging it in shows it is a solution. 38:47
Two integral curves also cannot touch, even tangentially. The lecturer says this is much less obvious than the crossing rule, but you still have to know it. 39:44

Words to know

isocline: A line where every short slope marker (line element) has the same slope. Drawing one helps you see the pattern of slopes.
integral curve: A curve whose slope at every point matches the slope markers drawn there. It is a picture of one solution.
line element: A tiny line segment drawn at a point to show the slope a solution must have at that point.
tangent: When two curves just touch at a point and share the same slope there, without crossing over.

Example

Think of water flowing down a hillside, with arrows showing which way it moves at each spot. If you drop a leaf into a narrow valley where all the arrows point downhill toward the same stream, the leaf cannot climb back out. The lecturer's pitcher plant is a similar trap, where the hairs all point down.

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Part 5 · 40:00–48:52

This part explains why solution curves of a differential equation cannot touch or cross, what the existence and uniqueness theorem promises, and why a worked example seems to break it at one spot.

Key ideas

Two solution curves cannot touch. If they met at a point, they would have to be the same curve from there on, so they cannot simply join up. 40:23
The Existence and Uniqueness Theorem says that through a point (x zero, y zero), the equation y' = f(x, y) has at least one solution, and only one. 41:31
The theorem has conditions. f(x, y) must be continuous near the point, and its partial derivative with respect to y must also be continuous there. The second condition is what guarantees uniqueness. 42:57
In the example, the solutions are lines that all pass through (0, 1). Points on the y-axis (except (0, 1)) have no solution, and (0, 1) has many. The theorem is not broken, because the equation's right side is undefined when x is 0, so the conditions fail there. 48:02

Words to know

Integral curve: A curve that solves the differential equation. Each point on it follows the slope rule the equation gives.
Existence: Whether a solution exists at all through a given point.
Uniqueness: Whether there is only one solution through a given point, with no other solution possible there.
Standard form: The usual way of writing a differential equation with dy/dx alone on one side, so the theorem's conditions can be checked.

Example

Imagine two hikers who each follow the same trail map. If the map is clear, each step leads to one clear next step, so the two hikers can never meet on the trail and then split up. In the same way, two solution curves cannot meet at a point and then go separate ways.

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Lecture 3: Solving First-order Linear ODEs →

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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