In this section, we introduced the concept of the forward Laplace transform and derived some common Laplace transforms that we will use throughtout this chapter. The following points summarize the essential concepts from the forward Laplace transform section:
Common Forms: A table of common Laplace transforms is provided, which doubles as a reference for inverse transforms. The focus is on recognizing forms that match the table entries for functions like \(\sin(bt), \cos(bt)\text{,}\) and others.
Direct Computation: When the function of \(s\) directly matches a form in the common Laplace transform table, the inverse Laplace transform can be easily computed.
Modifying Functions: When a function doesnβt match a known form, minor modifications, such as multiplying by missing constants or splitting fractions, can help.
Completing the Square: When dealing with quadratic expressions in the denominator, especially when the discriminant is negative, completing the square can transform the expression into a form that matches known inverse Laplace transforms. Several examples demonstrate this technique.
Partial Fraction Decomposition: For more complex rational functions, partial fraction decomposition breaks down the function into simpler fractions that match the common transform forms.
Completing the square is an essential technique for transforming quadratic expressions that donβt directly match a form in the table of common Laplace transforms. However, itβs not the only strategy available. In this section, weβll explore another important technique: partial fraction decomposition. This method is useful for breaking down complex fractions into simpler components that can each be matched with forms in the Laplace transform table.
When we want to take the inverse Laplace transform of a rational function with a second-degree polynomial in the denominator, we may complete the square or we may do a partial fraction decomposition. How will we know which is appropriate? Here are a few guidelines for you to consider.
Does the denominator factor in an obvious way? If so, factor the denominator and do a partial fraction decomposition if necessary.
If instead you end up with subtraction outside the parentheses, as in \((s - a)^2 - b^2,\) then you should factor and do a partial fraction decomposition. You may consider using the quadratic formula if the factorization is not obvious to you.
