What Are Functions Called That Are Linearly Dependent With Their Derivatives?

[Gaussian97]

TL;DR Summary

Is there a name for functions that are linearly dependent with its derivatives?

Is there a name for functions that are linearly dependent with its derivatives? i.e. a function $f(x)$ such that, for some value of $n$ it fulfills
$$f^{(n+1)} = \sum_{k=0}^{n} \alpha_k f^{(k)}$$
are linearly dependent?

 

[anuttarasammyak]

As an example
$$f(x)=e^{ax}$$as
$$f^{(n+1)}(x)=a f^{(n)}(x)$$so
$$\alpha_n=a$$ otherwise $$\alpha_k=0$$ There are other expressions, e.g.,
$$\alpha_k=\frac{1}{n+1}a^{n-k+1}$$
The component functions are all the same.

 

[Gaussian97]

Yes, if we define the set $D_k$ as the set of all such functions with $n=k-1$, then any exponential would be on $D_1$ because the first derivative is LD to the function itself, $\sin{x}$ and $\cos{x}$ would be on $D_2$ because the second derivative is LD to the function, and a polynomial with degree $j$ would be on $D_{j+1}$. But I was asking if there is a special name given to such functions.

 

[Office_Shredder]

I don't think you get polynomials in this set, the top degree gets destroyed by derivatives so you can never get rid of it with a linear equation in the derivatives.

I don't know of a name for these. I suspect they are all secretly linear combinations of the exponential function with complex arguments, e.g.

$$\sin(x)=\frac{ e^{ix} - e^{-ix}}{2}.$$

 

[Stephen Tashi]

Summary:: Is there a name for functions that are linearly dependent with its derivatives?

Is this equivalent to asking about all functions that are solutions to some homogeneous linear differential equation with constant coefficients?