Lecture notes · Differential Equations · Lecture 21

Using Laplace Transform to Solve ODEs with Discontinuous Inputs

MIT 18.03 Differential Equations, lecture 21, explained in plain English

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Part 1 · 0:00–10:00

This part introduces special step-shaped functions and shows why the Laplace transform can't tell apart functions that differ only for negative time.

Key ideas

The Laplace transform is useful because it handles functions that jump suddenly, which are called jump discontinuities. 0:29
The unit step function u(t) is zero before time zero and one after it. Its value exactly at zero is left undefined. Shifting it to jump at a point a gives u(t - a). 0:54
A unit box is a pulse that is one between a and b and zero elsewhere. It can be written as u(t - a) minus u(t - b), so it is built from step functions. 4:02
Multiplying any function by a unit box keeps only the part between a and b and wipes out the rest. 5:43
The Laplace transform only uses time from zero onward, so different functions that match for positive time can have the same transform. This means the inverse transform is not always unique. 9:41

Words to know

Laplace transform: A math tool that changes a function of time into a new function of s, which often makes equations easier to solve.
jump discontinuity: A spot where a graph suddenly jumps from one value to another, with no smooth link between them.
unit step function: A simple switch-like function that is zero before a set point and one after it.
inverse Laplace transform: The reverse process: starting from the transformed function and finding the original function of time.

Example

Think of a light switch that is off until you flip it on at a set time. The switch's on-or-off record only tells you what happens after you flip it, so it does not tell you what the light was doing before that. In the same way, the Laplace transform only looks at time from zero onward.

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Part 2 · 10:00–20:00

This part explains why shifting a function in time needs a special formula, and what that formula is.

Key ideas

The Laplace transform is for problems that start at time zero and go forward. It does not use anything that happened before time zero. 10:15
To make the function unique, the lecturer sets it to zero for negative times by multiplying it by the unit step function. This keeps the function the same for positive times. 11:37
You cannot write the Laplace transform of a shifted function using only the transform of the original. The part that moves past the vertical axis was lost when the transform was taken. 15:18
The correct formula uses the shifted function with the part before the shift wiped out. Its transform equals e to the minus as times the transform of the original function. 19:00
Real problems will not be written in the form f of t minus a. You can still use the formula after a simple change to the symbols. 19:41

Words to know

Laplace transform: A math tool that turns a function of time into a new function, so that some problems become easier to solve. It only looks at time zero and later.
unit step function: A switch-like function that equals one for positive times and zero for negative times. Multiplying by it makes a function zero before time zero.
translate: To slide a graph sideways along the time axis without changing its shape. Here the lecturer slides the graph to the right by a distance a.
inverse Laplace transform: The reverse step that turns a transformed function back into the original function of time.

Example

Think of a movie that starts at 8 PM. If you only care about what happens from 8 PM onward, you can ignore the earlier scenes. Shifting the movie to start later means you must still have the opening scenes, or the start will be wrong.

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Part 3 · 20:00–30:00

This part shows why shifting a function along the time axis by a delay a multiplies its Laplace transform by e to the minus as, and how to prove it.

Key ideas

The Laplace transform is a way of turning a function of t into a function of s. The formula covered here says that if you shift a function along the t-axis by a, its new transform is found by multiplying the old transform by e to the minus as. 20:31
This t-axis shift is the mirror image of the earlier formula, where the shift happens on the s-axis and the change is a multiplication by an exponential in t. Each formula moves on one side and multiplies on the other. 22:02
To prove the formula, the lecturer swaps in a new variable t1 = t minus a. This is a trick called changing the variable in a definite integral, and it changes the limits of the integral too. 23:03
The step function u is zero for negative values, so the part of the integral from negative a to zero adds nothing. That lets the lower limit become zero, which gives the Laplace transform of f. 25:34
A dummy variable is a letter used inside an integral that gets integrated away. Its name does not matter, so the result is still the Laplace transform of f. For a function like t squared, you write t as (t minus a) plus a to get it into the right form. 27:01

Words to know

Laplace transform: A tool that changes a function of time into a new function of s, which often makes equations easier to solve.
dummy variable: A placeholder letter inside an integral. It gets used up by the integral, so you can rename it and get the same answer.
definite integral: An integral with set start and end points, which gives a single number as the answer.
translation: Sliding a function sideways along an axis, without changing its shape.

Example

Think of a movie that starts at 7:00 pm. If the cinema moves the start to 7:30 pm, the movie is the same, just delayed by half an hour. In the same way, shifting a function by a delay a keeps its shape and only moves it along the time axis.

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Part 4 · 30:00–40:00

This part shows how to handle functions that switch on at a set time, using the Laplace transform and its inverse, with worked examples.

Key ideas

To find the Laplace transform of a shifted piece, change every t in the function to t plus the shift amount a. Then multiply by e to the minus as. 30:13
The unit step function u(t - a) is zero before time a and one after it. Its Laplace transform is e to the minus as times 1 over s. 32:05
For a function like u(t - 1) times t squared, you change t to t + 1 inside the function, expand it, and take the transform term by term. 34:05
When an exponential factor appears in an inverse transform, the answer is a discontinuous function. So you must write u(t) sine t, not just sine t, and handle each exponential piece separately. 39:00

Words to know

Laplace transform: A tool that changes a function of time into a function of a new variable s, which often makes equations easier to solve.
unit step function: A switch-like function that is zero before a set time and one after it. It is written u(t).
inverse Laplace transform: The reverse step: turning a Laplace answer back into a function of time.
discontinuous: Having a sudden jump or break, so the function is not smooth at one point.

Example

Think of a faucet that is turned on at a set moment. Before that moment the water is zero, and after it the water flows. The unit step function acts like that switch, so the formula can describe the flow starting at a set time.

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

Part 5 · 40:00–44:09

This part shows how to turn a two-piece answer with a switch into a clean cases form, using sine waves that start at time pi.

Key ideas

Multiplying by e to the minus pi s shifts a result to the right by pi. In time, the matching piece is the old answer with t replaced by t minus pi, and it switches on at t = pi. 40:00
The full answer is the sum of two pieces: u(t) times sine t, plus u(t minus pi) times sine of (t minus pi). The lesson says this form is correct but will lose credit. 40:43
To make the answer clear, split time into cases. Before pi, the second piece is zero, so only sine t is left. 41:24
After pi, the second piece turns on. Shifting the sine curve right by pi turns sine t into negative sine t, so the two pieces cancel and the answer is zero. 42:43
The final answer is sine t for t between 0 and pi, and zero for t greater than or equal to pi. 43:58

Words to know

Inverse Laplace transform: A way to turn a result in the s-world back into a normal function of time t.
Unit step u(t): A switch that is zero before a set time and one after it. The lesson uses u(t) as one for positive t and zero before.
Shift (translate): Sliding a graph left or right along the time line without changing its shape.
Cases format: Writing a function in separate parts, one for each stretch of time, so each stretch is easy to read.

Example

Think of a sprinkler that turns on at 3 o'clock and follows the same wave pattern from then on. Before 3, you only see the first pattern. After 3, the second pattern starts, and you describe what happens in each time period separately.

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

← Lecture 19: Introduction to the Laplace Transform · Lecture 25: Homogeneous Linear Systems 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.

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