Laplace transform has been used to solve Initial Value Problems. This is a topic covered in a standard differential equation course at most universities. In this post I will describe a strange situation: the solution of 2nd order ODE obtained by using Laplace transform appears to violate one of the two initial conditions. Then a short justification will be given. It turns out to be a very simple problem, just one may get caught off guard.
Background. Consider a 2nd order linear constant coefficient differential equation:
, , .
Applying Laplace transform, we get
, where the capital-case functions are the Laplace transforms of the corresponding lowercase functions: and . It follows that
, or equally, .
Applying the inverse transform, we obtain the solution of the IVP.
Laplace Transform of the delta functions. Let be the delta function at . We know that . In particular, .
A problematic example. Consider the IVP , , .
Laplace Transform: , or equally, .
Inverse Transform: .
The above process is fairly straightforward. Most of us would stop here and move onto the next problem. Some of you may double check:
(1) for , where the delta function also vanishes;
(2) , the first initial condition checked;
(3) , , wait, what? — it violates the second initial condition.
You can check the above computation. There is no error. How is it a solution since it does not satisfy the initial conditions that we started with? Should we call it a solution? Is Laplace transform not applicable here? These were my initial reactions.
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