Difference Equation

A difference equation is a mathematical equation that defines a sequence by relating each term to one or more previous terms. It is a discrete analogue of a differential equation, which relates the values of a function to its derivatives.

The given difference equation is:

yt+1 - 5yt = 1

This equation relates the current term yt+1 to the previous term yt. The goal is to find a solution for yt that satisfies the equation.

Solving the Difference Equation

To solve the given difference equation, we will follow these steps:

1. Find the homogeneous solution: In the homogeneous case, the right-hand side of the equation is zero. So we have:

yt+1 - 5yt = 0

To solve this homogeneous equation, we assume a solution of the form yt = r^t. Substituting this into the equation, we get:

r^(t+1) - 5r^t = 0

Simplifying the equation, we can divide both sides by r^t:

r - 5 = 0

This gives us the solution r = 5.

2. Find the particular solution: In the particular case, we have a non-zero right-hand side of the equation. So we have:

yt+1 - 5yt = 1

To find the particular solution, we assume a solution of the form yt = A, where A is a constant. Substituting this into the equation, we get:

A - 5A = 1

Simplifying the equation, we have:

-4A = 1

Dividing both sides by -4, we get:

A = -1/4

3. Find the general solution: The general solution is the sum of the homogeneous and particular solutions. So we have:

yt = general homogeneous solution + particular solution
= C * 5^t + (-1/4)

Where C is a constant determined by the initial condition.

Initial Condition

The initial condition given is yo = 7/4. Substituting this value into the general solution, we get:

7/4 = C * 5^0 + (-1/4)
7/4 = C - 1/4
C = 2

Final Solution

Substituting the value of C back into the general solution, we have:

yt = 2 * 5^t - 1/4

Therefore, the solution to the first-order difference equation yt+1 - 5yt = 1, with the initial condition yo = 7/4, is given by:

yt = 2 * 5^t - 1/4