# Thread: System of first order differential equations

1. ## System of first order differential equations

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!

2. Originally Posted by magodiafano
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

3. Originally Posted by Haven
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?

4. could someone help me?? it's very important!!