Homogeneous linear differential equation with constant coefficients

Definition

A homogeneous linear differential equation with constant coefficients, which can also be thought of as a linear differential equation that is simultaneously an autonomous differential equation, is a differential equation of the form:

$y^{(k)}+p_{k-1}{y}^{(k-1)}+\dots +p_1{y}^{\prime }+p_{0}y=0$

where $p_0,p_1,\dots ,p_{k-1}$ are all constants (i.e., real numbers).

Solution method

Consider the following polynomial:

$t^k+p_{k-1}{t}^{k-1}+\dots +p_{1}t+p_0=0$

This polynomial is called the characteristic polynomial of the differential equation. We consider various cases:

Case	Solution in that case
The polynomial has pairwise distinct real roots ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k$	The solution space has basis $e^{{\alpha }_{1}x},e^{{\alpha }_{2}x},\dots ,e^{{\alpha }_{k}x}$. In other words, the general solution is $C_1{e}^{{\alpha }_{1}x}+C_2{e}^{{\alpha }_{2}x}+\dots +C_k{e}^{{\alpha }_{k}x}$ where $C_1,C_2,\dots ,C_k$ are freely varying real parameters.
The polynomial splits completely into linear factors over the reals, but with possible repetitions. ${\alpha }_1$ occurs $s_1$ times, ${\alpha }_2$ occurs $s_2$ times, and so on till ${\alpha }_r$, which occurs $s_r$ times. We have $s_1+s_2+\dots +s_r=k$.	The solution space has basis all functions of the form $x^d{e}^{{\alpha }_{i}x}$ where $0\le d<s_i$ with $d$ an integer. Thus, for each $i$, there are $s_i$ basis vectors corresponding to ${\alpha }_i$. We get a total of $k$ basis vectors.
The polynomial splits completely over the complex numbers into distinct linear factors, but some of the roots are not real	For any real root $\alpha$, use $e^{\alpha x}$ as a basis vector. Non-real roots occur in complex conjugate pairs. For a pair $a\pm ib$, choose the vectors $e^{ax}\cos (bx)$ and $e^{ax}\sin (bx)$. Combining, we get a basis of $k$ vectors.
The general case	For a real root $\alpha$ of multiplicity $s$, the $s$ basis vectors are $x^d{e}^{\alpha x},0\le d<s$. For a pair of complex conjugates $a\pm ib$ of multiplicity $s$, the $2s$ basis vectors are $x^d{e}^{ax}\cos (bx),0\le d<s$ and $x^d{e}^{ax}\sin (bx),0\le d<s$.