Lecture notes · Differential Equations · Lecture 18

Introduction to the Laplace Transform

MIT 18.03 Differential Equations, lecture 18, explained in plain English

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Part 1 · 0:00–10:00

The lecturer explains where the Laplace transform comes from by starting with power series and turning a discrete sum into a continuous integral.

Key ideas

The Laplace transform is a popular tool engineers use to solve differential equations, and this lecture promises to explain where it comes from. 0:08
A power series adds up terms like a_n times x to the n. You can think of the list of coefficients a_n as a function that takes whole numbers and gives back real numbers. 1:44
A power series only adds up to a useful answer for some values of x. For 1 + x + x squared + ..., the answer 1 over (1 minus x) works only when x is between negative one and one. 3:39
To make the sum continuous, replace the sum over whole numbers with an integral over a continuous variable t. Integration is the continuous version of adding up many small pieces. 5:54
The lecturer picks the base e because it is the base people most like to differentiate and integrate. He also wants x to be a positive number between zero and one so that the integral has a chance to converge. 7:42

Words to know

power series: A long sum of terms, each made of a number times x raised to a whole-number power, like 1 + x + x squared + x cubed and so on.
converge: A sum or integral settles down to one definite number instead of growing without limit.
improper integral: An integral that runs out to infinity, which needs careful handling to see whether it settles to a number.
discrete: Made of separate, countable steps, like whole numbers 0, 1, 2, 3. The lecturer contrasts this with continuous, which means smooth and covering every value in between.

Example

Think of a shopping list with items you add up one at a time. Replacing the list with a smooth dial that you slide along, and adding up all the tiny amounts as you slide, is the same kind of change the lecturer makes when he swaps a sum for an integral.

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Part 2 · 10:00–20:00

This part introduces the Laplace transform, a way to turn a function of time t into a function of a new variable s, and starts working out the transform of the constant function one.

Key ideas

The speaker swaps the variable so the numbers are easier to work with. He sets log x equal to negative s, so s stays positive. 10:15
The Laplace transform takes a function of t and gives back a function of s. This is the continuous version of a power series sum. 12:05
A transform changes the variable, while an operator like differentiation keeps the same variable. This is why it is called a transform. 12:42
The transform is linear. The transform of a sum is the sum of the transforms, and constants can be pulled outside. This works because the definite integral is itself linear. 15:36
Because the integral runs to infinity, it is an improper integral. You find its value by taking a limit as R goes to infinity. 19:10

Words to know

Laplace transform: A rule that turns a function of t into a new function of s, using an integral from zero to infinity.
Operator: A rule that takes in a function and gives back a function, usually keeping the same variable, such as differentiation.
Linear: Behaves nicely with addition and multiplication by a constant. Doubling the input doubles the output, and sums split apart.
Improper integral: An integral with an endless limit, such as infinity. You solve it by checking what happens as the limit gets larger and larger.

Example

Think of a recipe that turns a list of ingredients into a finished dish. The input and output are different things, just as a Laplace transform turns a function of t into a function of s. The speaker says the transform works like a translator that changes the language, not just the wording.

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Part 3 · 20:00–30:00

This part shows that the Laplace transform of 1 is 1/s only when s is positive, and it derives a shortcut called the exponential shift for multiplying a function by e to the at.

Key ideas

Working out the integral of 1 gives 1/s, but only after taking a limit as R goes to infinity. The lecturer then notices a catch in that step. 20:43
The answer 1/s holds only when s is positive. If s is negative, the exponent becomes positive and grows without limit, so the integral does not settle down. 22:00
The lecturer says the Laplace transform is usually calculated only for certain kinds of functions, such as exponentials, sines and cosines, and polynomials. 24:01
If you multiply a function by e to the at, the new transform is the old transform with s replaced by s minus a. This is the exponential shift. 27:23
The shifted formula is valid only when s is greater than a, because the original transform is only good for values of s above a certain point. 28:42

Words to know

Laplace transform: A special integral that turns a function of time t into a function of a new variable s, so some problems become easier to solve.
kernel: The part of the integral that multiplies the function. In the Laplace transform it is e to the minus s t.
exponential shift: The rule that multiplying a function by e to the at turns the transform's s into s minus a.
limit: The value a expression gets closer and closer to as a variable grows toward some number, such as infinity.

Example

Think of a sliding ruler. If you slide the scale over by a, every mark now reads as a little less than before, which is like replacing s with s minus a. The shift only makes sense for the marks that are still on the ruler, just as the formula only works when s is greater than a.

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Part 4 · 30:00–40:00

This part shows how to find Laplace transforms of e to the a t, sines and cosines, and why you need partial fractions to turn results back into normal functions of time.

Key ideas

The exponential shift rule says that multiplying a function by e to the a t changes s to s minus a in its Laplace transform. For example, the transform of e to the a t is 1 over (s minus a). 30:37
The same rule works when a is a complex number, which lets you handle sines and cosines using complex exponentials. The lecturer uses the backwards Euler formula to write cosine as a sum of two exponentials. 32:23
The Laplace transform of cosine a t comes out to s divided by (s squared plus a squared). The lecturer says the sine version, a divided by (s squared plus a squared), is left for the recitation. 35:31
Solving differential equations takes three steps: transform the problem, solve it in s-land, then convert back. The hardest step is the last one, so you need inverse Laplace transforms. 36:29
Tables alone are not enough for inverse Laplace transforms. You rewrite the expression with partial fractions, then use the cover-up method to find each coefficient, and then look up each simple piece in the tables. 38:27

Words to know

Laplace transform: A way of changing a function of time into a new function of a variable s, which makes some hard equations easier to solve.
Exponential shift rule: A rule that says multiplying by e to the a t moves s over by a in the transform, so s becomes s minus a.
Partial fractions: Splitting one complicated fraction into several simpler fractions that are easier to work with and look up.
Inverse Laplace transform: Going backward from the s-version of a function to the original version in terms of time t.

Example

Think of a translation app. The exponential shift rule is like a fixed rule for turning one word into another, so once you know the rule, you can translate any phrase that uses that word. Partial fractions are like breaking a long sentence into short phrases you already know how to translate.

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Part 5 · 40:00–47:49

This part shows how to find the Laplace transform of t to the n, by using integration by parts over and over until the answer is clear.

Key ideas

The Laplace transform is linear, so a polynomial can be handled one term at a time. The only job left is finding the transform of a single power of t. 40:40
To find the transform of t to the n, use integration by parts. Treat t to the n as the part to differentiate and e to the negative st as the part to integrate, because it is easy to integrate. 41:22
At the upper limit, the exponential term must go to zero. That only happens if s is positive, so the lecturer restricts the calculation to positive s. 43:43
In the race to infinity, the exponential always beats any power of t. You can see this by applying L'Hopital's rule (a rule for limits) again and again. 44:41
Each step gives a smaller version of the same problem, so repeating the steps leads to the final answer: the Laplace transform of t to the n is n factorial over s to the n plus one. 47:18

Words to know

linear: A rule that lets you split a problem into pieces, solve each piece, and add the answers back together.
integration by parts: A method for integrating a product of two things by choosing one part to differentiate and the other to integrate.
parameter: A value that stays fixed during a calculation, like s here. It is treated as a constant while you work with t.
L'Hopital's rule: A trick for finding a limit when you get a fraction that goes to infinity over infinity. You differentiate the top and bottom and check again.

Example

Think of a race between a slow runner and a rocket. Even if the slow runner starts with a head start, the rocket always wins in the end. In this lesson, the exponential term is the rocket and the power of t is the slow runner.

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

← Lecture 15: Introduction to Fourier Series · Lecture 22: Using Laplace Transform to Solve ODEs with Discontinuous Inputs →

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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