Inverse Laplace Transforms
To solve the differential equation, we must eventually return to the time domain. This requires some algebra gymnastics.
Introduction
The Inverse Laplace Transform is denoted:
Unlike the forward transform (which is integration), the inverse transform usually just involves matching patterns from our table. The hard part is massaging to look like the table patterns.
Linearity
Just like the forward transform, the inverse is linear:
Partial Fractions
Most transforms result in rational functions . We split them up.
- Distinct Linear Factors: .
Results in . - Repeated Linear Factors: .
Results in . - Irreducible Quadratics: or .
Results in .
(Often requires Completing the Square).
Visual: Pole Locations
Interactive: Poles dictate Behavior
The denominator roots ("poles") in the s-domain determine the exponents and frequencies in the time domain.
Worked Examples
Example 1: Basic Linear
Find .
1. Factor Denominator:
.
2. Partial Fractions:
.
.
Set .
Set .
3. Transform Back:
.
(Note: This is also ).
Example 2: Repeated Roots
Find .
1. Setup:
.
.
2. Solve Coefficients:
Set .
Match 's' terms: .
3. Transform:
.
: Recall . Shift gives .
.
Example 3: Completing the Square
Find .
1. Complete Square:
.
This is a sine/cosine shifted by with .
2. Match Numerator:
We need terms.
.
.
3. Transform:
First term: .
Second term: Needs a 5 on top. .
.
Practice Quiz
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