Definition:Reduced Equation of Linear ODE with Constant Coefficients

Definition

Consider the linear $n$th order ODE with constant coefficients:

$(1): \quad \ds \sum_{k \mathop = 0}^n a_k \dfrac {\d^k y} {d x^k} = \map R x$

The equation:

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

is the reduced equation of $(1)$.

First Order Linear ODE

Consider the linear first order ODE with constant coefficients:

$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$

The equation:

$\dfrac {\d y} {\d x} + a y = 0$

is the reduced equation of $(1)$.

Second Order Linear ODE

Consider the linear second order ODE with constant coefficients:

$(1): \quad \dfrac {\d^2 y} {\d x^2} + p \dfrac {\d y} {\d x} + q y = \map R x$

The equation:

$\dfrac {\d^2 y} {\d x^2} + p \dfrac {\d y} {\d x} + q y = 0$

is the reduced equation of $(1)$.

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