# Inverse laplace transforms

## Main Question or Discussion Point

I am looking for inverse Laplace transform techniques with repeated roots. Example (10-4s)/(s-2)^2

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2 Exp[2t](-2+t)

You need Mathematica :tongue:

Here is a technique.

1. Write the equation setting the rational expression, in this case (10-4s)/(s-2)^2, equal to its partial fraction expansion (on the right hand side) with unknowns.

2. Multiply both sides by the denominator of the rational expression.

3. Expand the right hand side to be a polynomial in s, and do this to the left side too if necessary.

4. You know the polynomial on the left side is equal to the polynomial on the right hand side. So the cofficients must be equal. For example the coiefficent of s^2 on the left must be equal to the coefficient of s^2 on the right. So you now can write a system of linear equations and solve for the unknowns in the partial fraction expansion.

For an example see http://www.swarthmore.edu/NatSci/ec...on/PartialFraction.html#Repeated_Real_Roots." in the section titled "Example: Cross Multiplication"

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