8. The general solution of a linear system.

Consider the linear dynamical system


that is


We look for solutions on the form


If we instert these to the system we get


that is


This equation system has nontrivial solutions if and only if


that is, if and only if


This is the characteristic equation of the system and the solutions  its eigenvalues. If we now for each eigenvalues   solve the equation system


we obtain the eigenvectors


Then we have the general solution of our original dynamical system as


that is


where C1 and C2 are arbitrary constants.

Example 8:
 Assume that the matrix A is


This matrix has characteristic equation


with the eigenvalues 


and corresponding eigenvectors


This means that the general solution of the corresponding dynamical system is


that is


Example 9: Solve the system


Solution: We consider the matrix


and its characteristic equation


that is


The eigenvalues thus are


with corresponding eigenvectors


This means that the dynamical system has the general solution


that is


These are all complex solutions. We are actually only interested in the realsolutions. With help of Euler’s formula we get


If we now pick arbitrary real constants D1 and D2 and put


we get the general real solution