Section 10.2 Preparing the Backward Transform
So far, we have explored the forward (initial step) and backward (final step) Laplace transforms, as illustrated in the figures below. 
However, the transition between these two steps often requires some algebraic manipulation before the backward transform can be applied, as shown in the following figure. 
After solving for \(Y\) in Step 2a, the goal is to express \(Y\) as a sum of simpler components whose inverse Laplace transforms can be easily identified using the right-hand column of the Laplace Transform Table. This may involve algebraic techniques such as splitting rational functions, completing the square, or partial fraction decomposition. These methods will be covered in this section.
