Lecture notes · Differential Equations · Lecture 3

Solving First-order Linear ODEs

MIT 18.03 Differential Equations, lecture 3, explained in plain English

Want this lecture to stick? In LessonBound you get checked practice questions for every part, can ask about anything you don't get, and can listen with your screen off. Free, no ads.

Learn this lecture free

Watch or read the original lecture: MIT OpenCourseWare.

Part 1 · 0:00–10:00

This part introduces the first-order linear equation, explains why it is called linear, and says it is the model the lecture will use for the temperature concentration problem.

Key ideas

A first-order linear equation has a y-prime term, a y term, and a right-hand side, where each coefficient is a function of x. The independent variable is often x, but in real problems it is often time, t. 1:34
It is called linear because it is linear in y and y-prime, the way a high school linear equation is linear in its variables. 2:18
If the right-hand side is zero, the equation is called homogeneous, just as in 18.02. 3:25
The lecturer picks one standard form for this course: the coefficient of y-prime is 1, written as y-prime plus p(x) times y equals q(x). Getting the sign of p wrong leads to answers with totally different behavior. 6:26
This equation can always be solved, and it shows up in many models, such as temperature, mixing, radioactive decay, bank interest, and motion. Today's model is the temperature concentration model. 7:29

Words to know

independent variable: The variable you choose freely, like x or time t. The solution depends on it.
linear: Built from y and y-prime in a simple way, with no powers or products of them, like a straight-line equation.
homogeneous: The case where the right-hand side is zero. The lecturer says this is called homogeneous.
standard form: One agreed way of writing the equation, so every equation looks the same and can be solved the same way.

Example

Think of a cup of hot tea cooling on a table. How hot it is at each moment depends on time, and the rate it cools depends on how far it is from the room temperature. A first-order linear equation can describe this kind of change, which is why the lecturer says it shows up so often in models.

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

Part 2 · 10:00–20:00

This part shows how one first-order differential equation can model both heat moving through a metal box and salt moving across a membrane.

Key ideas

The lecturer renames the temperature model the conduction model, because it only looks at heat moving by conduction. The same math also works for salt concentration, which he calls the diffusion model. 10:21
The rate at which the inside temperature changes is proportional to the difference between the outside and inside temperatures. Proportional means one number goes up or down in step with the other. 14:05
The constant k must be positive. If the outside is hotter, the inside temperature should rise, so the rate of change has to come out positive. 15:11
The diffusion model uses the same equation, but C now stands for salt concentration inside and Ce for salt concentration outside. 16:38
Both situations are modeled by the same first-order differential equation, so the lecturer does not have to say which one he is following. 19:34

Words to know

conductivity: The constant k in the model. It measures how easily heat or salt gets through, and it is treated as a fixed number.
differential equation: An equation that describes how something changes over time, by linking a quantity to its rate of change.
proportional: Two things are proportional when one changes in step with the other, usually by being multiplied by a fixed number.
initial condition: A starting fact about the system, such as the temperature at the start time, that tells you which specific solution to use.

Example

Think of a hot cup of tea sitting in a cool room. The bigger the gap between the tea and the room, the faster the tea loses heat. The change depends on the difference, just as the model says. (This example is an illustration added for understanding, not something the lecture covers.)

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

Part 3 · 20:00–30:00

This part shows how to solve a first-order linear equation by multiplying it by a helper function called an integrating factor, so the left side can be integrated directly.

Key ideas

A first-order linear equation can be written in standard form as Y prime plus P of X times Y equals Q of X. The temperature equation can be rewritten to fit this form. 20:41
The temperature model still counts as linear even when the conductivity K changes over time, because the lecturer allows K and the outside temperature to vary. 21:46
An integrating factor U of X is a helper function. You multiply the whole equation by it, and you choose it so the left side becomes the derivative of U times Y. 25:29
The product rule shows that U must satisfy U prime equals P times U. Solving that by separating variables gives U as e to the power of the integral of P. 27:19
You only need one working U, so no arbitrary constant is needed. Once the left side is a derivative, you solve by integrating both sides with respect to X. 29:07

Words to know

standard linear form: A neat way of writing a first-order linear equation, as Y prime plus P times Y equals Q, so the parts are easy to compare.
integrating factor: A helper function you multiply an equation by, chosen so the left side can be integrated easily.
product rule: A calculus rule for finding the derivative of two functions multiplied together.
separate variables: Moving all the Y or U terms to one side and all the X terms to the other so each side can be integrated.

Example

Think of a bath that is filling while also draining. The integrating factor is like a clever tool that lets you see the whole change as one simple step you can add up. Once you have that tool, you can find the answer just by adding up the pieces.

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

Part 4 · 30:00–40:00

This part shows the steps for solving a first-order linear equation by using an integrating factor, and warns about the mistakes people often make.

Key ideas

First, rewrite the equation in standard linear form, which looks like Y prime plus P times Y equals Q. Skipping this step is the first common mistake. 30:00
The integrating factor is e raised to the integral of P. If P has the wrong sign, the answer will be wrong, and often badly wrong. 30:14
Multiply both sides of the standard form by the integrating factor, then integrate. The left side should then be a simple derivative that you can integrate directly. 31:22
Check your work by differentiating the left side with the product rule to confirm it matches the original left side. 34:21
Multiply the right side Q by the integrating factor too. Forgetting this is the most common mistake, according to the lecturer. 37:00

Words to know

standard linear form: The neat layout of the equation, with the Y prime term first, then P times Y, then Q on the other side. You must get the equation into this layout before you start.
integrating factor: A special multiplier, found as e to the integral of P, that you multiply the whole equation by so the left side becomes easy to integrate.
product rule: The rule for taking the derivative of two things multiplied together. The lecturer uses it to check that the left side is correct.
integral: The opposite of a derivative. It adds up the pieces of a function, and you use it to find the original function.

Example

Think of following a recipe. You must add the ingredients in the right order and in the right amounts, and if you forget one, like the salt on the right side of the equation, the dish comes out wrong. Solving this kind of equation works the same way: each step has to be done in order.

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

Part 5 · 40:00–50:24

This part shows how to solve a first-order linear equation with a constant coefficient, using a temperature cooling model, and what the solution looks like over a long time.

Key ideas

Multiplying an equation by the right helper term can turn the left side into the derivative of one product. Once that happens, you can undo the derivative by integrating. 40:22
A starting value, called an initial condition, lets you find the constant in the solution. For example, if Y of zero is 1, the constant C comes out to 2. 41:55
In the temperature model, the outside temperature term must have the same constant k in front of it as the temperature term. Otherwise the units will not match. 44:07
For this model, the helper term is e to the kt. Multiplying by it lets you integrate both sides and solve for the temperature. 45:13
The solution has two parts. The steady state is what remains as time goes to infinity. The transient part fades away, so the starting value stops mattering in the long run. 49:35

Words to know

integrating factor: A helper term you multiply both sides of an equation by, so the left side becomes the derivative of one product that you can then integrate.
indefinite integral: An integral with no starting or ending limits. Its answer includes an unknown constant, so it stands for a whole family of functions.
steady state: The part of the solution that is left over as time goes on forever. It does not fade away.
transient: The part of the solution that fades away as time goes on. It depends on the starting condition, but it disappears in the long run.

Example

Think of a cup of hot coffee sitting in a cool room. At first its temperature depends on how hot it started, but after a long time it settles to room temperature no matter what the starting temperature was. The fading part is the transient, and the room temperature is the steady state.

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

← Lecture 1: The Geometrical View of y'= f(x,y) · Lecture 9: Solving Second-order Linear ODE's with Constant Coefficients →

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.

LessonBound · Free lecture notes · Privacy · Terms · New York