# 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 