Section 22.11
Laplace for Systems
We can transform the entire system into the s-domain, turning it into a system of linear algebra equations.
1
Introduction
Laplace transform is linear, so it works on vectors too.
2
The Method
Step-by-Step
- Transform: Apply Laplace to each equation.
- Solve: You now have a system of algebraic equations for and . Solve it (Cramer's Rule or Substitution).
- Inverse: Use partial fractions to find and .
3
Worked Examples
Example 1: Initial Value Problem
Solve , with .
1. Transform:
.
.
2. Rearrange:
.
.
3. Substitute:
.
.
4. Inverse:
Partial Frac: .
.
Find y: or just .
Example 2: Double Tank
Coupled tanks with initial salt.
Laplace is often easier than eigenvalues for IVPs because it handles the initial conditions automatically.
You don't need to find at the end.
Example 3: Discontinuous Force
.
Laplace is much better here than Variation of Parameters.
Simply add to the algebraic system.
4
Practice Quiz
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