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This question already has an answer here:

We know that $Ce^x$ and $0$ are the two functions whose first derivative is equal to itself, but what about derivatives of a higher order? For example, the second derivative of $e^{-x}$ is equal to itself, but not the first, and the fourth derivative of $sin(x)$ is equal to itself.

In short, are there other examples of functions whose nth derivative is equal to itself, where $n>1$?

Thank you kindly!

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marked as duplicate by J. M. is a poor mathematician, Community Jun 15 '16 at 1:07

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The second derivative of $ce^x+ke^{-x}$ is the same as the original function, same goes for $\cosh(x)$ and $\sinh(x)$.

The third derivative of $e^{\omega x}$ is the same as the original function.

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