Definition:Linear nth Order ODE with Constant Coefficients

Definition

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

A linear $n$th order ODE with constant coefficient is an ordinary differential equation of order $n$ which can be manipulated into the form:

$\ds \sum_{k \mathop = 0}^n a_k \dfrac {\d^k y} {d x^k} = \map R x$

where $a_k$ is a real constant for all $0 \le k \le n$.

That is:

$a_n \dfrac {\d^n y} {d x^n} + a_{n - 1} \dfrac {\d^{n - 1} y} {d x^{n - 1} } + \dotsb + a_1 \dfrac {\d y} {d x} + a_0 y = \map R x$

Also presented as

Such an equation can also be presented in the form:

$a_n y^{\paren n} + a_{n - 1} y^{\paren {n - 1} } + \dotsb + a_1 y' + a_0 y = \map R x$

or:

$\paren {a_n D^n + a_{n - 1} D^{n - 1} + \dotsb + a_1 D + a_0} y = \map R x$

Also see

• Solution of Constant Coefficient Linear nth Order ODE

Sources

• 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 3$. Equations of higher order and systems of first order equations: $\S 3.1$ The $n$th order equation