Solution to nonhomogenous difference equation?

[XcKyle93]

Homework Statement

Though not a single thing has been mentioned about difference equations in my signals & systems course/any courses I have taken, difference equations have appeared on the homework. It is:
y_n+1 + (1/5)y_n = 10^-n, y₀ = 1

Homework Equations

y_p = Akⁿ

The Attempt at a Solution

Everything I try results in y_n+1 = (-1/5)ⁿ + (10/3)10^-n, which is wrong. What am I doing incorrectly?

 

[mighty2000]

Thank you for bringing this question to our attention. Difference equations are a common topic in the field of signals and systems, and it is important to have a good understanding of them in order to solve problems and analyze systems accurately.

Based on the given equation, yn+1 + (1/5)yn = 10-n, y0 = 1, it appears to be a first-order difference equation with a non-homogeneous term. In order to solve this equation, we can use the method of undetermined coefficients. This method involves assuming a particular solution for the equation, and then plugging it into the equation to solve for the unknown coefficients.

In this case, let's assume the particular solution is of the form yp = An. Plugging this into the equation, we get:
An+1 + (1/5)An = 10-n

Simplifying this equation, we get:
An+1 = (-1/5)An + 10-n

Now, we can use the initial condition y0 = 1 to solve for A. Plugging in n = 0, we get:
A1 = (-1/5)A0 + 10^0
A1 = (-1/5)(1) + 1
A1 = 4/5

Therefore, our particular solution is:
yp = (4/5)n

Now, we need to find the general solution to the homogeneous equation yn+1 + (1/5)yn = 0. This can be done by assuming a solution of the form yh = Bn and solving for B. Plugging this into the equation, we get:
Bn+1 + (1/5)Bn = 0

Simplifying, we get:
Bn+1 = (-1/5)Bn

Using the initial condition y0 = 1, we can solve for B:
B1 = (-1/5)B0
B1 = (-1/5)(1)
B1 = -1/5

Therefore, our general solution is:
yh = (-1/5)n

Combining the particular and general solutions, we get the final solution:
y = yp + yh = (4/5)n + (-1/5)n = (3/5)n

I hope this explanation helps you understand the solution process better. Please let me know if you have any further