Hi, I am having trouble with the following problem. I dont know where I am doing it wrong... please help me out...

**Problem:**
Apply the eigenvalue method to find the particular solution to the system of differential equations

which satifies the initial conditions

___________________

**Attempt:**
$\displaystyle {A} = \begin{pmatrix}

{6}&{7}\\

{5}&{8}

\end{pmatrix}\\

=> \left|A-\lambda I \right| = \begin{pmatrix}

{6-\lambda}&{7}\\

{5}&{8-\lambda}

\end{pmatrix} = 0\\

=> \lambda = 13 \quad{or} \lambda = 1 \\

$

So I found the eigenvalues and solved for eigenvector.

So for lambda = 13 eigenvector $\displaystyle \begin{pmatrix}

{1}\\

{1}

\end{pmatrix}$

for lambda = 1 eigenvector $\displaystyle \begin{pmatrix}

{1}\\

{-5/7}

\end{pmatrix}$

$\displaystyle x(t) = {C_1}\begin{pmatrix}

{1}\\

{1}

\end{pmatrix}e^{13t}+{C_2}\begin{pmatrix}

{1}\\

{-5/7}

\end{pmatrix}e^{1t}$

Now for the given initial conditions I finally got:

x1 = -16/3*e^(13t) - 7/3*e^(t)

x2 = -16/3*e^(13t) + 5/3*e^(t)

But its wrong...

Please help...

Many Thanks....