Section 21.7
IVPs with Step Functions
The true power of Laplace Transforms: handling systems that switch on and off without needing to break the problem into pieces.
1
Introduction
In Method of Undetermined Coefficients, a discontinuous requires splitting the problem into time intervals and matching 's at the boundary.
Laplace does it all in one shot.
2
Strategy
1
Express g(t) with Steps
Use to write the forcing function as a single line equation.
2
Transform & Solve
Isolate . Expect terms like .
3
Inverse with Shift
Recall .
3
Worked Examples
Example 1: Turned On Force
Solve , .
1. Transform:
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2. Partial Fractions for H(s):
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3. Shift Back:
Multiply by and shift .
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Example 2: Pulse Input
Solve , .
1. Transform:
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2. Base Inverse:
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3. Result:
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Example 3: Interpretation
In Example 2, the system stays at 0 until .
It charges up towards 1 (exponential approach).
At , the source cuts off, and the system discharges (decays back to 0).
4
Practice Quiz
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