Properties and Formulas
We don't need to integrate every time. This section builds our "Table of Transforms" using powerful properties.
Introduction
Just like differentiation, the Laplace Transform has rules that allow us to combine simple functions to handle complex ones.
Linearity
Theorem
Since integration is linear, the Laplace Transform is linear:
This means we can transform term by term.
First Translation Theorem
Concept
Multiplying by an exponential in the time domain causes a shift in the s-domain.
This is crucial for solving DEs with exponential forcing or damping.
Interactive: Shifting S
The pole (vertical asymptote) moves from to .
Derivatives of Transforms
What if we multiply by instead of ?
Multiplying by in time corresponds to differentiating (and flipping sign) in s-domain.
Worked Examples
Example 1: Linearity
Find .
1. Separate Terms:
.
2. Use Table:
.
.
3. Combine:
.
Example 2: Translation
Find .
1. Identify Parts:
, shift is .
.
2. Apply Shift:
Replace every with .
.
.
Example 3: Derivative of Transform
Find .
1. Identify Formula:
.
2. Differentiate:
.
.
3. Combine signs:
.
Practice Quiz
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