The way Laplace transforms, as presented in a typical differential equation course, are not very useful. Laplace transforms are useful, but not as presented.
The use of Laplace transforms is presented is as follows:
- Transform your differential equation into an algebraic equation.
- Solve the algebraic equation.
- Invert the transform to obtain your solution.
This is correct, but step 3 is typically presented in a misleading way. For pedagogical reasons, students are only given problems for which the last step is easy. They’re given a table with functions on the left and transforms on the right, and you compute an inverse transform by recognizing the result of step 2 in the right column.
Because of the limitations listed above, Laplace transforms, as presented in an introductory course, can only solve problems that could just as easily be solved by other methods presented in the same course.
What good is it, in an undergraduate classroom setting, if you reduce a problem to inverting a Laplace transform but the inverse problem doesn’t have a simple solution?
Of course in practice, rather than in a classroom, it might be very useful to reduce a complicated problem to the problem of inverting a Laplace transform. The latter problem may not be trivial, but it’s a standard problem. You could ask someone to solve the inversion problem who does not understand where the transform of the solution came from.
Laplace inversion theorem
The most well-known Laplace inversion theorem states that if f is a function and F is the Laplace transform of f, then you can recover f from F via the following integral.
It’s understandable that you wouldn’t want to present this to most differential equation students. It’s not even clear what the right hand side means, much less how you would calculate it. As for what it means, it says you can calculate the integral along any line parallel to the imaginary axis. In practice, the integral may be evaluated using contour integration, in particular using the so-called Bromwich contour.
It might be difficult to invert the Laplace transform, either numerically or analytically, but at least this is a separate problem from whatever led to this. Maybe the original problem was more difficult, such as a complicated delay differential equation.
Post-Widder theorem
There is a lesser-known theorem for inverting a Laplace transform, the Post-Widder theorem. It says
where F(n) is the nth derivative of F. This may not be an improvement—it might be much worse than evaluating the integral above—but it’s an option. It doesn’t involve functions of a complex variable, so in that sense it is more elementary [1].
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[1] The use of the word elementary in mathematics can be puzzling. Particularly in the context of number theory, elementary essentially means “without using complex variables.” An elementary proof may be far more difficult to follow than a proof using complex variables.
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