Linear Algebra with Differential Equations/Non-Linear Differential Equations
Some Graphical Analysis
So far we've dealt with being a constant matrix, and other niceties; but when it is otherwise, and thus a non-linear differential equation, the best way to find a solution is by graphical means. By taking the independent variables on the axis of a graph, we can note several types of behavior that suggest the form of a solution.
So without adue, here are the main types of behaviors, and their suggested causes:
A nodal source (the graph tends away from a point): real, distinct positive eigenvalues.
A nodal sink (the graph approaches in towards a point): real, dinstinct negative eigenvalues.
A saddle point (the graph approaches from one end and deviates away at another): real, disntinct, opposite eigenvalues.
A spiral point (spirals in or away from a point): a complex eignevalues.
A series of ellipses around a point: a purely imaginary eigenvalue.
A star point (straight lines deviating or coming towards a point): repeated eigenvalues.