1. Phase planes for linear systems with real eigenvalues
SECTION 3.3

2. Summary
Suppose Y(t) = etV is a straight-line solution to a system of DEs.
What happens to Y(t) = (x(t), y(t)) as t ∞?
How does your answer depend on the sign of ?
Suppose dY/dt = AY has two real eigenvalues. Its general solution is Y(t) = k1etV1 + k2etV2.
If both eigenvalues are negative, then the exponential terms go to 0 as t approaches infinity. The origin is the only equilibrium and it is a sink.
If both eigenvalues are negative, then the exponential terms go to infinity as t approaches infinity. The origin is the only equilibrium and it is a source.
If the eigenvalues are mixed, then the behavior is more complicated… It’s called a saddle.

3. Exercises
p. 287: 1, 3, 5
p. 288: 17, 18 (use the DiffEq software)