Science and engineering math: Difference equation, non-homogeneous

• chatterbug219
In summary: Diff Eq: an+2 - 5an+1 + 6an = 4nSolved for an = A(3)nIn summary, Homework Statement I was trying to solve for an equation when I got two solutions. One for when it is homogeneous, and one for when it isn't. I don't know where to start for non-homogeneous.
chatterbug219

Homework Statement

Solve the difference equation: an+2 - 5an+1 + 6an = 4n
Subject to a0 = 0 & a1 = 1

an = Arn

The Attempt at a Solution

I got two solutions for the first part for when it is homogeneous by substituting an = Arn into the equation and solving for r.
an = A(3)n
an = A(2)n

I just don't know where to start for non-homogeneous. I tried an = B*4n and I got B*4n+2 -5B*4n+1 + 6B*4n = 4n but I don't know where to go from there.

welcome to pf!

hi chatterbug219! welcome to pf!
chatterbug219 said:
I tried an = B*4n and I got B*4n+2 -5B*4n+1 + 6B*4n = 4n but I don't know where to go from there.

erm

4n+2/4n = … ?

So B = 1/2
And plugging that in...
an = (1/2)(4n) for the particular solution
Making the general solution:
an = A(2)n + (1/2)(4n)
Right?

hi chatterbug219!

(just got up :zzz:)
chatterbug219 said:
… Making the general solution:
an = A(2)n + (1/2)(4n)
Right?

you mean an = A(2)n + B(3)n + (1/2)(4n)

(or you could write it 2n(2n-1 + A) + 3nB )

No...if you plug in the initial conditions they final answers don't match up...
Only 2 works

yes, but to solve for the initial condition, you also need to find A

chatterbug219 said:

Homework Statement

Solve the difference equation: an+2 - 5an+1 + 6an = 4n
Subject to a0 = 0 & a1 = 1

an = Arn

The Attempt at a Solution

I got two solutions for the first part for when it is homogeneous by substituting an = Arn into the equation and solving for r.
an = A(3)n
an = A(2)n

I just don't know where to start for non-homogeneous. I tried an = B*4n and I got B*4n+2 -5B*4n+1 + 6B*4n = 4n but I don't know where to go from there.

The simplest way is to determine the generating function $A(z) = \sum_{n=0}^{\infty} a_n z^n$, then invert it to find $\{ a_n \}.$ Google "difference equations" to see hundreds of examples.

RGV

1. What is the difference between a difference equation and a non-homogeneous equation?

A difference equation is a mathematical equation that describes the relationship between the current value of a variable and its past values, while a non-homogeneous equation is a type of difference equation that includes a constant or non-constant term on the right side of the equation.

2. How are difference equations and non-homogeneous equations used in science and engineering?

Both difference equations and non-homogeneous equations are used to model and analyze dynamic systems in science and engineering. They are commonly used in fields such as physics, biology, economics, and control engineering to describe the behavior of systems over time.

3. What is the purpose of using difference equations and non-homogeneous equations?

The purpose of using these equations is to understand and predict the behavior of complex systems by breaking them down into smaller, simpler equations. This allows for easier analysis and manipulation of the system, and can provide insights into the system's stability and response to different inputs.

4. Can difference equations and non-homogeneous equations be solved analytically?

In general, non-homogeneous equations cannot be solved analytically and require numerical methods for solution. However, some special cases of non-homogeneous equations, such as linear equations, can be solved analytically using techniques such as the method of undetermined coefficients.

5. What are some real-world applications of difference equations and non-homogeneous equations?

Difference equations and non-homogeneous equations have a wide range of applications in science and engineering. They are used to model population growth, chemical reactions, economic systems, and many other dynamic systems. They are also commonly used in control systems and signal processing to regulate and manipulate the behavior of systems.

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