Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form
$$\sum_{i=1}^m c_it^{n_i}e^{\lambda_it}\quad(c_i\neq 0)$$
where $m, n_1,\ldots, n_m\ge0$ are arbitrary natural numbers and $\lambda_1,\ldots ,\lambda_m$ arbitrary complex numbers. The precise statement is this: Let $\mathcal F$ be the set of all exponential polynomials. Then for any linear $n\times n$ system $\vec x\prime(t)=A \vec x (t)$ with arbitrary initial value $\vec x (0)\in \mathbb C^n$ we have $\vec x(t)\in\mathcal F^n$. The set $\mathcal F$ has many other important properties, which are useful e.g. for solving boundry problems. One of the most important properties is that they can always be integrated by
$$\int x^ke^{\lambda t}= {(-1)^{k+1}k! \over \lambda ^{k+1}}+\sum_{i=0}^k{(-1)^ik^{\underline i}\over \lambda^{i+1}}x^{k-i}e^{\lambda x} \quad (\lambda \neq0)\tag{*}$$
$$\int x^k= {x^{k+1}\over k+1},$$
where $k^{\underline i}$ denotes the falling factorial $k(k-1)\cdots(k-i+1)$ and the integral $\int f$ is defined as $\int_0^x f(\mathcal E)d\mathcal E$. Prove (*) by induction on k.