Nonconstant Coefficients
What if the coefficients are functions of t? We can use the derivative property in reverse.
Introduction
Standard Laplace methods work well for constant coefficients (e.g., ).
But if we have , we can't just replace with .
The Derivative Property
Recall:
This lets us handle terms like or .
However, this often transforms an ODE in into a differential equation in !
Worked Examples
Example 1: Bessel-like Equation
Solve , .
1. Transform Terms:
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2. Assemble Equation:
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3. Solve s-DE:
Standard linear DE for . Separation of variables usually works here.
This demonstrates complexity, but simplification leads to .
4. Inverse:
.
Example 2: Laguerre Equation (Order 1)
Solve .
1. Transform:
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2. Combine:
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3. Separate Variables:
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4. Inverse:
Partial fractions gives .
Example 3: Simple Case
Solve , ?
Warning:
If , this has no solution at (singularity).
1. Transform:
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2. Solve:
.
3. Inverse:
is the derivative of the Delta function (distribution), not a normal function.
This shows limitations: classic solutions might require distribution theory.
Practice Quiz
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