Example 5.  Use the matrix exponential to find the general solution for the system of  D.E.'s   [Graphics:Images/MatrixExponentialMod_gr_210.gif],  where
    [Graphics:Images/MatrixExponentialMod_gr_211.gif].  

Solution 5.

[Graphics:../Images/MatrixExponentialMod_gr_212.gif]


[Graphics:../Images/MatrixExponentialMod_gr_213.gif]

 

 

A fundamental matrix solution is   [Graphics:../Images/MatrixExponentialMod_gr_214.gif].

[Graphics:../Images/MatrixExponentialMod_gr_215.gif]


[Graphics:../Images/MatrixExponentialMod_gr_216.gif]

 

 

The matrix exponential is  [Graphics:../Images/MatrixExponentialMod_gr_217.gif].

[Graphics:../Images/MatrixExponentialMod_gr_218.gif]


[Graphics:../Images/MatrixExponentialMod_gr_219.gif]

 

 

The solution to the D. E. with initial conditions  [Graphics:../Images/MatrixExponentialMod_gr_220.gif],   is  [Graphics:../Images/MatrixExponentialMod_gr_221.gif].  

[Graphics:../Images/MatrixExponentialMod_gr_222.gif]


[Graphics:../Images/MatrixExponentialMod_gr_223.gif]

 

 

[Graphics:../Images/MatrixExponentialMod_gr_224.gif]


[Graphics:../Images/MatrixExponentialMod_gr_225.gif]

[Graphics:../Images/MatrixExponentialMod_gr_226.gif]

We are done.

Aside.  We compare the above work, with the construction of  [Graphics:../Images/MatrixExponentialMod_gr_227.gif]  using Mathematica's subroutine  MatrixExp.

[Graphics:../Images/MatrixExponentialMod_gr_228.gif]


[Graphics:../Images/MatrixExponentialMod_gr_229.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004