Example 2.  Use the matrix exponential to find the general solution for the system of D. E.'s   
        [Graphics:Images/MatrixExponentialMod_gr_149.gif]    

Solution 2.

First, write the system in vector and matrix form  [Graphics:../Images/MatrixExponentialMod_gr_150.gif].

[Graphics:../Images/MatrixExponentialMod_gr_151.gif]


[Graphics:../Images/MatrixExponentialMod_gr_152.gif]

 

 

A fundamental matrix solution is   [Graphics:../Images/MatrixExponentialMod_gr_153.gif].

[Graphics:../Images/MatrixExponentialMod_gr_154.gif]


[Graphics:../Images/MatrixExponentialMod_gr_155.gif]

 

 

The matrix exponential is  [Graphics:../Images/MatrixExponentialMod_gr_156.gif].

[Graphics:../Images/MatrixExponentialMod_gr_157.gif]


[Graphics:../Images/MatrixExponentialMod_gr_158.gif]

 

 

The solution to the D. E. with initial conditions  [Graphics:../Images/MatrixExponentialMod_gr_159.gif],   is  [Graphics:../Images/MatrixExponentialMod_gr_160.gif].  

[Graphics:../Images/MatrixExponentialMod_gr_161.gif]


[Graphics:../Images/MatrixExponentialMod_gr_162.gif]

 

 

[Graphics:../Images/MatrixExponentialMod_gr_163.gif]


[Graphics:../Images/MatrixExponentialMod_gr_164.gif]


We are done.

Aside.  We compare the above work, with the construction of  [Graphics:../Images/MatrixExponentialMod_gr_166.gif]  using Mathematica's subroutine  MatrixExp.

[Graphics:../Images/MatrixExponentialMod_gr_167.gif]


[Graphics:../Images/MatrixExponentialMod_gr_168.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004