# Solve Difference Equation | General Solution | No Particular Solution

• haoku
In summary, the conversation is about finding the general solution to a difference equation given by T(a+2)-7T(a+1) +6T(a)= 6f. The auxiliary equation has been found to be A(1)^n+B(6)^n, but there is difficulty in finding the particular solution. It is suggested to try a polynomial in a for the solution.
haoku

## Homework Statement

Given difference equation
T(a+2)-7T(a+1) +6T(a)= 6f
Find the general solution of this equation

## Homework Equations

I have found the auxiliary equation be
A(1)^n+B(6)^n
But seems can't find the particular solution of that question.
Is it impossible to do this question?

## The Attempt at a Solution

Hi haoku!
haoku said:
T(a+2)-7T(a+1) +6T(a)= 6f

What's f?

f is a constant? Then "trying" a constant as solution won't work because a constant already satisfies the homogeneous equation (n0= 1). Try T(n)= Cn as a solution and determine C.

Sorry this is 6a

haoku said:
Sorry this is 6a

ah!

in that case, try a polynomial in a.

## What is a difference equation?

A difference equation is a mathematical equation that describes the relationship between a sequence of values. It is used to predict the value of the next term in a sequence based on the previous terms.

## How do you solve a difference equation?

To solve a difference equation, you first need to identify the type of difference equation it is (linear, nonlinear, homogeneous, nonhomogeneous, etc.). Then, you can use various techniques such as substitution, iteration, or generating functions to find the general solution.

## What is a general solution?

A general solution is the most basic form of a solution to a difference equation. It includes all possible solutions that satisfy the equation. It is usually expressed in terms of arbitrary constants and can be used to find a particular solution for a specific set of initial conditions.

## What is a particular solution?

A particular solution is a specific solution to a difference equation that satisfies both the equation and a given set of initial conditions. It can be obtained by substituting the initial conditions into the general solution and solving for the arbitrary constants.

## Why might a difference equation not have a particular solution?

A difference equation may not have a particular solution if the initial conditions do not satisfy the equation. In some cases, the equation may also have no solution at all. This can happen if the equation is inconsistent or if the general solution involves complex numbers and the initial conditions are real numbers.

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