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Without using Picard's theorem for existence and uniqueness of solutions of ordinary differential equations, if we solve a differential equation with the method of the Laplace transform, do we get uniqueness?

My argument is that as there is uniqueness of the Laplace transform, and when we find the solution of the corresponding algebraic solution we find it's unique solution, maybe we get the only solution of the ODE when we apply the inverse Laplace transform back.

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1 Answer 1

maybe we get the only solution of the ODE when we apply the inverse Laplace transform back.

And maybe we don't, who knows. Everything is easy up to and including the step when you find $Y(s)$, the function you get from solving the algebraic equation. But now you have to answer two questions:

  1. does there exist a function $y$ such that $\mathcal{L}\{y\}=Y$?
  2. is such $y$ unique?

When you work with a concrete function $Y$, you may be able to answer 1 by explicitly producing such $y$. You can also get a general existence statement by using the convolution property of $\mathcal{L}$. But uniqueness remains an issue even then: what if there is another $y$, not given by convolution? I don't think there is an easy proof of uniqueness.

If you do manage to prove uniqueness of $y$, you will indeed obtain an existence/uniqueness theorem -- for linear equations with constant coefficients only. The cost/benefit ratio is rather poor compared to Picard's theorem.

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