We call them linear non-homogeneous if the equation can be written in the form
and linear homogeneous if, in addition to being linear non-homogeneous,
The method of characteristic equations is for homogeneous equations and the methods of undetermined coefficients and of variation of parameters for homogeneous equations.
If the equation is linear homogeneous and further are constant, then the equation is referred to as a constant-coefficients equation:
and we can apply the method of characteristic equations to solve such an equation. Note that is assumed to be non-zero since we are working with a second order equation.
Consider a mass hanging at rest on the end of a vertical spring of length , spring constant and damping constant .
Let denote the displacement, in units of feet, from the equilibrium position. Note that since represents the amount of displacement from the spring's equilibrium position (the position obtained when the downward force of gravity is matched by the will of the spring to not allow the mass to stretch the spring further) then should increase downward. Then by Newton's Third Law one can obtain the equation
where is any external force, which for simplicity we will assume to be zero.