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]:

Where:

- The integral ∫ is an indefinite integral.
- F(t) is the objective function.
- f(s) is called the
*image*or*resultant function*.

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:

- Look the function up in a table,
- If the function isn’t in a table, then integrate (using the formula).
- 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 + e^{6t})^{2}?

Step 1: Look up the function f (t) = (3 + e^{6t})^{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:

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 + e
^{6t})^{2} - = 9 + 6e
^{6t}+ e^{12t}.

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

- 9
- 6 * e
^{6t} - e
^{12t}

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.

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**Stephanie Glen**. "Laplace Transform: Definition, How to Find" From

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