Lecture Notes in Differential Equations - Bruce E. Shapiro

382 LESSON 34. CRITICAL POINTS
This means that we could define new constants at any fixed time t
such that
a(t) = Ae λ1t and b(t) = Be λ2t (34.53)
a(t)v + b(t)w = 0 (34.54)
Since mathbfv and w are linearly independent, the only way this can happen
is when a(t) = 0 and b(t) = 0 at the same same time. There is no t
value for which this can happen because
e λt ≠ 0 (34.55)
for all possible values of λ. Thus the only way for (34.54) to be true is for
A = B = 0. This corresponds to the solution (x, y) = (0, 0), which is the
point at the origin.
Figure 34.3: Phase portraits typical of an unstable (left) and stable (right)
node.
slow eigendirection
fast eigendirection
fast eigendirection
slow eigendirection
Unstable Node
Stable Node
The geometry is illustrated in figure 34.3. The two straight lines passing
through the origin correspond to A = 0, B ≠ 0 and B = 0, A ≠ 0 respectively,
namely the two eigendirections. The solutions on the eigendirections
are Ae λ1t and Be λ 2 t, with different initial conditions represented by different
values of A and B. If the eigenvalues are positive, a particle that starts
along one of these eigendirections (i.e., has initial conditions that start the
system on an eigendirection) moves in a straight line away from the origin
as t → ∞ and towards the origin as t → −∞. If the eigenvalues are negative,
the particle moves in a straight line away from the origin as t → −∞
and towards the origin ast → ∞. Trajectories that pass through other
points have both A ≠ 0 and B ≠ 0 so that y = Ave λ1t + Bwe λ2t . If both
eigenvalues are positive, then for large positive time (as t → ∞) the fast