9.5 and 9.6 - Homogeneous Linear Systems with Constant Coefficients
Fact: Suppose the matrix has eigenvalues (some of them may be repeated) and linearly independent eigenvectors . Then the general solution to the homogeneous system is given by
where ‘s are arbitrary constants.
Fact: Suppose the real matrix has complex eigenvalues with corresponding eigenvectors then two linearly independent solutions to the homogeneous system are given by
5.4 - Phase Plane of Systems
For each of the following examples, we are given a matrix and its eigenvalues/eigenvectors.
Write the general solution to the system
Draw the phaseplane of the system and discuss the asymptotic behavior of the solution curves
Discuss the type and stability of the origin
Case 1. has two distinct real-valued eigenvalues of opposite signs. Case 3b. has complex eigenvalues with Case 3c. has complex eigenvalues with