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 , 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 :
- 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.
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:
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:
- 6 * e6t
The solution is:
Boyd, S. Table of Laplace Transforms. Retrieved May 9, 2021 from: https://web.stanford.edu/~boyd/ee102/laplace-table.pdf
 The L.Transform. Retrieved May 9, 2021 from: https://lpsa.swarthmore.edu/LaplaceXform/FwdLaplace/LaplaceXform.html
 Dyke, P. (1999). An Introduction to LT’s and Fourier Series. Springer London.
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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