Section 21.5
Solving IVPs with Laplace
We put it all together. Transform, Algebra, Inverse Transform. The "Three Step Program" for Differential Equations.
1
Introduction
Why use Laplace Transforms instead of ?
- It handles initial conditions automatically (no solving for at the end).
- It handles piecewise forcing functions (which happen all the time in engineering) elegantly.
2
The Workflow
1
Transform
Take of both sides. Use for .
2
Solve
It's just algebra now! Isolate on one side.
3
Inverse
Use Partial Fractions and Completing the Square to find .
3
Visual: Mapping
Interactive: Input to Output
Changing initial conditions (like velocity) just changes the numerator constants in .
4
Worked Examples
Example 1: First Order Linear
Solve , .
1. Transform:
.
.
2. Solve for Y:
.
3. Inverse:
Partial Fractions: .
.
. .
.
.
Example 2: Second Order w/ Initial Conditions
Solve , .
1. Transform LHS:
.
.
2. Transform RHS:
.
3. Isolate Y:
.
Note: .
.
4. Inverse:
.
(using formula).
.
Example 3: Undamped Harmonic
Solve , .
1. Transform:
.
. (ICs moved to RHS).
2. Solve for Y:
.
3. Inverse:
Standard form for sine: where .
.
.
5
Practice Quiz
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