Hi

i need help with this system of first order differential equations. Starting conditions are

\(\displaystyle x_{1} (0) =1\)

\(\displaystyle x_{2} (0) =1\)

The system is:

\(\displaystyle \frac{dx_{1}(t)}{dt}= -\frac{601}{400}x_{1}(t) + \frac{769}{1000}x_{2}(t) \)

\(\displaystyle \frac{dx_{2}(t)}{dt}= -\frac{201}{400}x_{1}(t) - \frac{1231}{1000}x_{2}(t) \)

Thank you!

The system becomes

\(\displaystyle \left[ \begin{matrix}\frac{dx_{1}(t)}{dt} \\ \frac{dx_{2}(t)}{dt}

\end{matrix} \right] \) =\(\displaystyle \left[ \begin{matrix} -\frac{601}{400} & \frac{769}{1000} \\ -\frac{201}{400} &- \frac{1231}{1000} \end{matrix} \right]\) \(\displaystyle \left[ \begin{matrix} x_1(t) \\ x_2(t) \end{matrix}\right]\)

Now we pull out the matrix:

\(\displaystyle A =\left[ \begin{matrix} -\frac{601}{400} & \frac{769}{1000} \\ -\frac{201}{400} &- \frac{1231}{1000} \end{matrix} \right]\)

And now we assume the form of the solution:

\(\displaystyle \left[ \begin{matrix} x_{1}(t) \\ x_{2}(t)

\end{matrix} \right] \) = \(\displaystyle e^{\lambda_1t}\eta_1 + e^{\lambda_2t}\eta_2\)

\(\displaystyle \lambda_1\) and \(\displaystyle \lambda_2\) are the eigenvalue of the matrix A.

while \(\displaystyle \eta_1\) and \(\displaystyle \eta_2\) are the corresponding eigenvectors