Solutions of Homogeneous Linear System of DE
where and
are eigenvalue and eigenvector for
respectively.
Superposition Principle
If and
are two solutions to a homogeneous SDE
, then
is also a solution for any scalars
,
.
Euler’s formula
General Solutions (Complex Eigenvalues)
1) Let be an eigenvalue corresponding to eigenvector
. (The eigenvectors are complex conjugates:
.)
2) Construct
3) The general solution is
How to Sketch Phase Portrait
Probably the best video on how to sketch Phase Portrait: