Define the system of equations;

$$x_{n+1} = x_n - y_n$$

$$y_{n+1} = 2x_n + 4y_n$$

so if we take $$A = \begin{pmatrix}

1 & -1\\

2 & 4

\end{pmatrix}$$

and define

$$\begin{pmatrix}

x_{n+1} \\

y_{n+1} \\

\end{pmatrix} = \begin{pmatrix}

1 & -1 \\

2 & 4 \\

\end{pmatrix}\begin{pmatrix}

x_n \\

y_n \\

\end{pmatrix}$$

Then given that the eigenvalues and a corresponding eigenvector to the eigenvalue can be defined as

$$\lambda_1 = 3$$

$$\mu_1 = \begin{pmatrix}

1 \\

-2 \\

\end{pmatrix}$$

$$\lambda_2 = 2$$

$$\mu_2 = \begin{pmatrix}

1 \\

-1

\end{pmatrix}$$

(I already checked this)

Is it then true that the general solution can be defined as

$$x_n = 3^nc_1 + 2^nc_2$$

$$y_n = -2*3^nc_1 - 2^nc_2$$

Reinout

reinout-g
Jun 2, 2014

#1**+5 **

Perhaps I've misunderstood, but if c1 and c2 are initial values (n=0) for x and y then (for n = 1):

These don't look the same to me!

Alan
Jun 2, 2014