Solution of homogeneous case
Characteristic equation and roots
Solving the homogeneous equation involves first solving its characteristic equation
for its characteristic roots i = 1, ..., n. These roots can be solved for algebraically if n ≤ 4, but not necessarily otherwise. If the solution is to be used numerically, all the roots of this characteristic equation can be found by numerical methods. However, for use in a theoretical context it may be that the only information required about the roots is whether any of them are greater than or equal to 1 in absolute value.
It may be that all the roots are real or instead there may be some that are complex numbers. In the latter case, all the complex roots come in complex conjugate pairs.
Solution with distinct characteristic roots
If no characteristic roots share the same value, the solution of the homogeneous linear difference equation can be written in terms of the characteristic roots as
where the coefficients ci can be found by invoking the initial conditions. Specifically, for each time period for which an iterate value is known, this value and its corresponding value of t can be substituted into the solution equation to obtain a linear equation in the n as-yet-unknown parameters; n such equations, one for each initial condition, can be solved simultaneously for the n parameter values. If all characteristic roots are real, then all the coefficient values ci will also be real; but with non-real complex roots, in general some of these coefficients will also be non-real.
Converting complex solution to trigonometric form
If there are complex roots, they come in pairs and so do the complex terms in the solution equation. If two of these complex terms are and the roots can be written as
where the non-subscript i is the imaginary unit and M is the modulus of the roots: Then the two complex terms in the solution equation can be written as
where is the angle whose cosine is and whose sine is the last equality here made use of de Moivre's formula.
Now the process of finding the coefficients and guarantees that they are also complex conjugates, which can be written as Using this in the last equation gives this expression for the two complex terms in the solution equation:
which can also be written as
where is the angle whose cosine is and whose sine is
Depending on the initial conditions, even with all roots real the iterates can experience a transitory tendency to go above and below the steady state value. But true cyclicality involves a permanent tendency to fluctuate, and this occurs if there is at least one pair of complex conjugate characteristic roots. This can be seen in the trigonometric form of their contribution to the solution equation, involving and
Solution with duplicate characteristic roots
In the second-order case, if the two roots are identical (), they can both be denoted as and a solution may be of the form
Conversion to homogeneous form
If b ≠ 0, the equation
is said to be non-homogeneous. To solve this equation it is convenient to convert it to homogeneous form, with no constant term. This is done by first finding the equation's steady state value—a value y * such that, if n successive iterates all had this value, so would all future values. This value is found by setting all values of y equal to y * in the difference equation, and solving, thus obtaining
assuming the denominator is not 0. If it is zero, the steady state does not exist.
Given the steady state, the difference equation can be rewritten in terms of deviations of the iterates from the steady state, as
which has no constant term, and which can be written more succinctly as
where x equals y–y *. This is the homogeneous form.
If there is no steady state, the difference equation
can be combined with its equivalent form
to obtain (by solving both for b)
in which like terms can be combined to give a homogeneous equation of one order higher than the original.