Calculus How To

Laplace Transform: Definition, How to Find

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The Laplace transform is an integral transform— a way to solve certain ordinary differential equation problems via a type of integration. The method “magically simplifies” challenging problems [1], turning them from differential equations into algebraic ones, which makes them easier to solve.

The most important property of the Laplace transform is linearity, which means we can add a series of results together to solve a more complicated one.

Definition of Laplace Transform

The Laplace transform f(s) is defined as [2]:
laplace transform definition

Where:

The Laplace Transform of a function can usually be looked up in a table, without any need to integrate. A table of Laplace transforms can be found here.

How to Find a Laplace Transform

The general steps to find any Laplace transform:

  1. Look the function up in a table,
  2. If the function isn’t in a table, then integrate (using the formula).
  3. If the function isn’t integrable then break the function into component parts, then look those parts up in a table.

Example Problem #1: What is the Laplace Transform of the constant function f (t) = 1?

Step 1: Look up the function f(t) = 1 in the table. The function is at the top of page 2.

The Laplace transform is 1/s.

That’s it!

Example Problem #2: What is the Laplace Transform of f (t) = (3 + e6t)2?

Step 1: Look up the function f (t) = (3 + e6t)2 in the table.
This function is not in the table, so let’s move on to Step 2.


Step 2: Insert the function into the Laplace transform formula:
using the laplace transform formula

It’s also not easily integrable in its current form (I used Symbolab’s calculator to confirm this). So we need to move on to Step 3.

Step 3: Break the function into parts (we can do this because of the property of linearity). Expanding the expression, we get:

  • f (t) = (3 + e6t)2
  • = 9 + 6e6t + e12t.

Step 2: Look up the function parts in the table. We have three parts to look up in this example:

  • 9
  • 6 * e6t
  • e12t

The solution is:

That’s it!

References

Boyd, S. Table of Laplace Transforms. Retrieved May 9, 2021 from: https://web.stanford.edu/~boyd/ee102/laplace-table.pdf
[1] The L.Transform. Retrieved May 9, 2021 from: https://lpsa.swarthmore.edu/LaplaceXform/FwdLaplace/LaplaceXform.html
[2] Dyke, P. (1999). An Introduction to LT’s and Fourier Series. Springer London.

CITE THIS AS:
Stephanie Glen. "Laplace Transform: Definition, How to Find" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/laplace-transform-definition-find/
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