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I have been stuck on this problem, I don't know where to start. The exact question is:
Determine the finite-dimensional spaces $W$ of differentiable functions with this property: If $f$ is in $W$ then $df/dx$ is in $W$.

There is a hint: "Review the solutions of a homogeneous, constant coefficient differential equation: $d^ny/dx^n+a_1*d^{n-1}y/dx^{n-1}+...+a_{n-1}*dy/dx+a_n=0$"

I know that if $f$ is in $W$ then all of its derivatives must be in $W$. I understand that the exponential functions clearly are in some W. But aren't the polynomial in some $W$ too since there is always a point at which their derivative is $0$? Extending that thought, there must be an infinite number of those $W$.. Can you please put me on the track?

BobTahri
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Yes, there are infinitely many spaces of this kind. For example, all polynomials of degree up to $n$ form such a space, for every $n\in\mathbb N$. This already gives you infinitely many spaces. But there are more! You can throw in any exponential function, $e^{\lambda x}$, and linear combinations of such functions. And trigonometric functions too.

Time to review the hint. If $f\in W$, then for $n=\dim W$ the functions $f,f',f'',\dots, f^{(n)}$ are linearly dependent. Therefore, $f$ satisfies an ODE of the form mentioned in the hint. Recall the general form of solutions of such ODE: they are spanned by $x^k e^{\alpha x} \cos\beta x$ and $x^k e^{\alpha x} \sin\beta x$, where $\alpha,\beta,k$ run over all triples such that $(\alpha+i\beta-\lambda)^{k+1}$ divides the characteristic polynomial $P(\lambda)$ associated with the ODE. So, all elements of $W$ are linear combinations of such monomial-exponential-trigonometric products.