System of first order differential equations

Apr 2010
9
0
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!
 
Jul 2009
197
32
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
 
Apr 2010
9
0
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
ehm i've several problem in computing autovectors... i used eig function in matlab but i'm not sure of the result! i'm not able to extract the eigenvector from the created matrix!
what shall i do?
I obtained this result

eigenvalues = -1.3668 + 0.6067i
-1.3668 - 0.6067i

while eigenvectors:

V =

0.7777 0.7777
0.1373 + 0.6135i 0.1373 - 0.6135i

how could i use this result in order to find the solution of the system?