# Thread: Solve The Difference Equation

1. ## Solve The Difference Equation

$\displaystyle f_{n+2} - 2f_{n+1}+ 5f_n = 0$

given that $\displaystyle f_0=1$ and $\displaystyle f_1=3$

Auxiliary equation:

$\displaystyle m^2 - 2m + 5 = 0$

Therefore:

$\displaystyle m = 1 + 2i$ or $\displaystyle m = 1 - 2i$

What would the complementry fuction be and paticular solution to work towards the solution of:

$\displaystyle f_n = [(1-i)/2][1+2i]^n + [(1-i)/2][1-2i]^n$

2. Originally Posted by louboutinlover
$\displaystyle f_{n+2} - 2f_{n+1}+ 5f_n = 0$

given that $\displaystyle f_0=1$ and $\displaystyle f_1=3$

Auxiliary equation:

$\displaystyle m^2 - 2m + 5 = 0$
Which is derived from an assumed solution of the form $\displaystyle f_n=m^n$

Therefore:

$\displaystyle m = 1 + 2i$ or $\displaystyle m = 1 - 2i$
OK, so you have a general solution of the form:

$\displaystyle f_n=A (1+2\text{i})^n+B(1-2\text{i})^n$

and the initial conditions will allow you to determine $\displaystyle A$ and $\displaystyle B$, which you appear to have done.

What would the complementry fuction be and paticular solution to work towards the solution of:

$\displaystyle f_n = [(1-i)/2][1+2i]^n + [(1-i)/2][1-2i]^n$

CB

3. So using the initial conditions would give:

$\displaystyle A+B=1$

$\displaystyle A(1+2i)+B(1-2i)=3$

But then how do you solve these equations ?

4. Originally Posted by louboutinlover
So using the initial conditions would give:

$\displaystyle A+B=1$

$\displaystyle A(1+2i)+B(1-2i)=3$

But then how do you solve these equations ?
$\displaystyle A=1-B$ from 1st,

$\displaystyle (1-B)(1+2\text{i})+B(1-2\text{i})=3$ by substituting from the above into the second.

So:

$\displaystyle B[-(1+2\text{i})+(1-2\text{i})]=3-(1+2\text{i})$

$\displaystyle B=\frac{-1+1\text{i}}{2\text{i}}=\frac{1+i}{2}$

Assuming my algebra is right (check it, it appears to differ from what you are give by a sign here or there).

CB