Using a table of Laplace transforms
You can use a table of Laplace transforms instead of the definition
To find the Laplace transform ???L\left\{f(t)\right\}??? of a function ???f(t)??? using a table of Laplace transforms, you’ll need to break ???f(t)??? apart into smaller functions that have matches in your table.
Using the formulas in the table to transform each of the smaller functions, you’ll then bring the transforms back together to generate the transform of the original function.
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An example of how to match transforms in a table to the differential equation
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Transforming the combination of an exponential, trigonometric, and power function
Example
Use a table of Laplace transforms to find the Laplace transform of the function.
???f(t)=e^{2t}-\sin{(4t)}+t^7???
To find the Laplace transform of a function using a table of Laplace transforms, you’ll need to break the function apart into smaller functions that have matches in your table.
We’ll look at each term separately and try to find a transform formula for ???e^{2t}???, ???\sin{(4t)}???, and ???t^7???.
For ???e^{2t}??? we’ll use the transform ???e^{at}=\frac{1}{s-a}??? and get
???e^{2t}=\frac{1}{s-2}???
For ???\sin{(4t)}??? we’ll use the transform ???\sin{(at)}=\frac{a}{s^2+a^2}??? and get
???\sin{(4t)}=\frac{4}{s^2+4^2}=\frac{4}{s^2+16}???
For ???t^7??? we’ll use the transform ???t^n=\frac{n!}{s^{n+1}}??? and get
???t^7=\frac{7!}{s^{7+1}}=\frac{7!}{s^8}???
Replacing the terms in the original function with their transforms, we get
???F(s)=\frac{1}{s-2}-\frac{4}{s^2+16}+\frac{7!}{s^8}???
This is the Laplace transform of ???f(t)=e^{2t}-\sin{(4t)}+t^7???.
