How to Find the General Solution of a Non-Homogeneous Recurrence Relation
This article explains the process of finding the general solution of a specific non-homogeneous recurrence relation: an - 6an-1 - 12an-2 - 8an-3 n22n for n ≥ 3, with initial conditions a0 1, a1 2, a2 1. The procedure involves solving the corresponding homogeneous recurrence relation and then finding a particular solution to the non-homogeneous one.
Step 1: Solving the Homogeneous Recurrence Relation
The first step is to solve the associated homogeneous recurrence relation:
an - 6an-1 - 12an-2 - 8an-3 0
The characteristic equation for this is:
r3 - 6r2 - 12r - 8 0
This factors as:
(r - 2)3 0
which gives us the triple root r 2. Therefore, the general solution to the homogeneous recurrence relation is:
an A2n Bn2n Cn22n
Step 2: Finding a Particular Solution for the Non-Homogeneous Recurrence Relation
To find a particular solution to the non-homogeneous recurrence relation, we need to consider the form of the right-hand side, which is n22n. Typically, for such a form, we would try a solution of the form:
an Dn52n En42n Fn32n
However, because Dn52n, En42n, and Fn32n are also solutions to the homogeneous recurrence, we need to multiply each by an additional factor of n to obtain a particular solution. Therefore, we try:
an Gn62n Hn52n In42n
Substituting this into the original recurrence relation and simplifying, we observe that the coefficients of n62n and n52n terms will cancel out, leaving only a coefficient for n42n. This simplifies to:
αn2 βn γ 0
where α, β, γ are linear expressions in G, H, I. Solving this system of equations, we find:
α 1, β 0, γ 0
Thus, a particular solution is:
an Gn62n Hn52n In42n
Setting α, β, γ to zero, we have:
an Gn62n Hn52n
Step 3: Combining the Solutions and Applying Initial Conditions
The general solution is given by the sum of the homogeneous and the particular solutions:
an A2n Bn2n Cn22n Gn62n Hn52n
Substituting the initial conditions a0 1, a1 2, a2 1, we get the following system of equations:
a0 A20 B020 C0220 G0620 H0520 1
a1 A21 B121 C1221 G1621 H1521 2
a2 A22 B222 C2222 G2622 H2522 1
Solving this system for the constants, we find the exact values that satisfy the equation.
Conclusion
The process of finding the general solution of the given recurrence relation involves solving the homogeneous equation, finding a particular solution using polynomial forms, and then applying initial conditions to determine the constants. By following these steps, one can systematically find the general solution to any non-homogeneous recurrence relation.