L(y_k) := a₀y_k+d + a₁y_k+d-1 + ... + a_dy_k = g_k

See Rosen 6/e section 7.2, esp. Theorems 4 and 6, or Introduction to Difference Equations p. 146.

Characteristic equation (auxiliary equation)

 a₀ r² + a₁r + a₂ = 0 if order/degree d = 2.

Roots of the characteristic equation

General solution of the homogeneous equation

• if the roots of the characteristic equation are distinct

   

• in the case of a repeated root

   

Particular solution of the nonhomogeneous equation: See below for trial solutions.

General solution of the nonhomogeneous equation

Solve the initial-condition (or boundary-condition) equations for the constants in the general solution.

Solution of the given nonhomogeneous equation satisfying the given initial conditions

Check: Compare values of y_k for the recurrence plus initial conditions with the values of your solution for k = 0, 1, 2, ....

Trial solutions for the nonhomogeneous equation L(y_k)=g_k

Right-hand term gk		Trial solution yk*
Geometric sequence, a rk		C rk
kn or polynomial in k of degree n		Polynomial in k of degree n: A0 + A1k + A2k2 + ... + Ankn
knrk or geometric times polynomial		Geometric times polynomial: rk( A0 + A1k + A2k2 + ... + Ankn )
sin(bk) or cos(bk) or linear combination thereof		C sin(bk) + D cos(bk)

Here the capital letters represent constants whose values are determined by requiring the trial solution to satisfy the nonhomogeneous equation.

Modification rules
When the trial solution has one or more terms in common with the general solution of the homogeneous equation, we must modify the trial solution by multiplying it by k^p where p is a just-high-enough integer power to make sure that the trial solution does not have any terms in common with the homogeneous solution. See Goldberg p. 146 for an example.