[DEFINITIONS] Second-order constant-coefficient equations

Constant-coefficient equation

A second-order constant-coefficient linear ODE has the form

ay''+by'+cy=f(x), \qquad a\neq0,

where $a$ , $b$ and $c$ are constants. It is homogeneous when $f(x)\equiv0$ .

The same idea extends to an $n$ th-order equation: each derivative has a constant coefficient and the corresponding characteristic polynomial has degree $n$ .

Characteristic equation

For the homogeneous equation

ay''+by'+cy=0,

the exponential trial function $y=e^{rx}$ gives the characteristic equation

ar^2+br+c=0.

The trial is useful because

y'=re^{rx},\qquad y''=r^2e^{rx},

so substitution gives

(ar^2+br+c)e^{rx}=0.

Since $e^{rx}\neq0$ , the differential equation reduces to an algebraic equation. For an $n$ th-order constant-coefficient equation, replace $y^{(j)}$ by $r^j$ to obtain the corresponding degree- $n$ polynomial.

Spring-mass-damper model

For a mass attached to a spring with spring constant $k$ and viscous friction coefficient $c$ , Newton's law gives

m\ddot{x}+c\dot{x}+kx=0.

The characteristic roots determine whether the motion is non-oscillatory or oscillatory and whether its amplitude grows or decays.

With an external force, the right side becomes a forcing term. The solution then has a transient homogeneous response and a particular forced response. In the usual mechanical units, $m$ is mass, $c$ has mass/time units and $k$ has mass/time $^2$ units.

[1]:Damped oscillation
[2]:Periodic forcing

Definitions from Second-order constant-coefficient equations

Constant-coefficient equation

Characteristic equation

Spring-mass-damper model

On this page