The Definition
We leave the time domain to solve differential equations algebraically in the s-domain.
Introduction
Solving differential equations directly (using characteristic equations, undetermined coefficients, etc.) can be messy, especially with discontinuous forcing functions (like a switch turning on/off).
The Laplace Transform is a tool that converts differential equations (hard) into algebraic equations (easy).
1. Transform the DE to the "s-domain".
2. Solve the algebra problem for .
3. Transform back to find .
The Definition
Improper Integral
Let be a function defined for . The Laplace transform is:
The result is a function of the new variable .
Visual: Why e^(-st)?
Interactive: Convergence Kernel
For the integral to converge (finite area), the term must decay faster than grows. This implies restrictions on .
Existence Conditions
The transform exists if:
- Piecewise Continuous: doesn't have infinite discontinuities.
- Exponential Order: . It doesn't grow faster than an exponential (like ).
Most physical functions (sines, polynomials, exponentials) meet these criteria.
Worked Examples
Example 1: Transform of a Constant
Find .
1. Definition:
.
2. Integrate:
.
.
3. Evaluate Limit:
If , .
.
Example 2: Transform of Exponential
Find .
1. Definition:
.
2. Integrate:
This is the same form as Example 1, but with instead of .
.
3. Conclusion:
Ideally .
.
Example 3: Transform of Sine
Find using Integration by Parts twice.
1. Setup:
.
Parts 1: .
2. Result after Parts:
After doing parts twice and solving the algebraic equation for (a classic Calculus II trick):
.
3. Note:
Similarly, .
Practice Quiz
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