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 real solutions. 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

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