Chapter Contents

• Laplace Transforms
• 1a. The Unit Step Function - Definition
• Oliver Heaviside
• 1b. The Unit Step Function - Products
• 2. Laplace Transform Definition
• Table of Laplace Transformations
• 3. Properties of Laplace Transform
• 4. Transform of Unit Step Functions
• 5. Transform of Periodic Functions
• 6. Transforms of Integrals
• 7. Inverse of the Laplace Transform
• 8. Using Inverse Laplace to Solve DEs
• 9. Integro-Differential Equations and Systems of DEs
• 10. Applications of Laplace Transform
• Laplace Transforms Problem Solver

• Comments, Questions?

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1b. Products Involving Unit Step Functions

When combined with other functions defined for `t > 0`, the unit step function "turns off" a portion of their graph.

The concept is related to having a switch in an electronic circuit open for a period of time (so there is no current flow), then the switch is closed (so the current begins to flow).

Example 1 - Products with Unit Functions

(a) If `f(t) = sin\ t`, then the graph of `g(t) = sin\ t · u(t − 2π)` is

The `sin\ t` portion starts at `t = 2π`, because we have multiplied `sin\ t` by `u(t − 2π)`.

We use the dot (`·`) for multiplication so that it is easier to read.

(b) If `f(t) = 10e^(-2t)`, then the graph of `g(t) = 10e^(-2t)· u(t − 5)` is

The portion `10e^(-2t)` starts at `t = 5`.

Product of u(t) vs. Shifting the Function Along the t-axis

Note the differences between the following:

`f(t) · u(t)`, where the `f(t)` part begins at `t = 0`.

`f(t) · u(t − a)`, where the `f(t)` part begins at `t = a`.

`f(t − a) · u(t)`, where the `f(t)` part has been shifted to the right by `a` units and begins at `t = 0`.

`f(t − a) · u(t − a)`, where the `f(t)` part has been shifted to the right by `a` units and begins at `t = a`.

Let's see some examples.

Example 2

Let `f(t) = 4t + 2` and `a = 1`. We see different combinations of shifting with different starting points.

(a) `g_1(t) = f(t) · u(t) = (4t + 2) · u(t)`

In this example, the `4t + 2` part starts at `t = 0`.

(b) `g_2(t) = f(t) · u(t − a) = (4t + 2) · u(t − 1)`

In this example, the `4t + 2` part starts at `t = 1`.

(c) `g_3(t) = f(t − a) · u(t) = (4(t − 1) + 2) · u(t) = (4t − 2) · u(t)`

In this example, the `4t + 2` part has been shifted 1 unit to the right and starts at `t = 0`.

(d) `g_4(t) = f(t − a) · u(t − a) = (4t − 2) · u(t − 1)`

In this example, the `4t + 2` part has been shifted 1 unit to the right (like example (c)) and starts at `t = 1`.

Example 3

Let `f(t) = sin\ t` and `a = 0.7` and we combine them to shift our graph and start at different times, similar to what we did in Example 1.

(a) `g_1(t) = sin\ t · u(t)`

In this example, the `sin\ t` part starts at `t= 0`.

(b) `g_2(t) = sin\ t · u(t − 0.7)`

In this example, the `sin\ t` part starts at `t = 0.7`.

(c) `g_3(t) = sin (t − 0.7) · u(t)`

In this example, the `sin\ t` part has been shifted `0.7` units to the right, and it starts at `t=0`.

(d) `g_4(t) = sin (t − 0.7) · u(t − 0.7)`

In this example, the `sin\ t` part has been shifted `0.7` units to the right, and it starts at `t = 0.7`.

Exercises

Rewrite the following functions in a suitable way and then sketch the functions:

1. `f(t) = u(t) + (1 − t) · u(t − 1)` `+ (t − 2) · u(t − 2)`

Answer

2. `f(t) = t^2 · u(t) − (t^2− 4) · u(t − 2)`

Answer

3. `f(t) = u(t) + (sin\ t − 1) · u(t − π/2) − (sin\ t + 1) · u(t − (3π)/2) + u(t − 2π)`

Answer

4. `f(t) = 3t^2· u(t)` `+ (12 − 3t^2) · u(t − 2)` ` + (4t − 40) · u(t − 4)` ` − 4(t − 7) · u(t − 7)`

Answer

Scientific Notebook Aside...

NOTE: To graph unit step functions using Scientific Notebook, we must realise that SNB recognises "Heaviside(t)", but not u(t).

So we need to define `u(t)` as `"Heaviside"(t)`, so SNB will graph it properly. Simply type `u(t) = "Heaviside"(t)`, and click on the "New definition" button. Nothing seems to happen, but if you click on the "Show Definitions" button you will see that it is defined. Now you can graph unit step functions in terms of `u(t)`.

Graph of `y = 3·u(t − 4)`