Lecture Notes in Differential Equations - Bruce E. Shapiro

387
Complex Conjugate Pair with nonzero real part.
If 0 < T 2 < 4∆ the eigenvalues will be a complex conjugate pair, with real
part T. T may be either positive or negative, but T = 0 is excluded from
this category (if T = 0, the eigenvalues will either be purely imaginary,
when∆ > 0, or real with opposite signs, if∆ < 0).
Writing
µ = T/2, ω 2 = 4∆ − T 2 (34.69)
the eigenvalues become λ = µ ± iω, µ, ω ∈ R. Designating the corresponding
eigenvectors as v and w, the solution is
where
y = e µt [ Ave iωt + Bwe −iωt] (34.70)
= e µt [Av (cos ωt + i sin ωt) + Bw (cos ωt − i sin ωt)] (34.71)
= e µt [p cos ωt + q sin ωt] (34.72)
p = Av + Bw (34.73)
q = i(Av − Bw) (34.74)
are purely real vectors (see exercise 6) for real initial conditions. The factor
in parenthesis in (34.70) gives closed periodic trajectories in the xy plane,
with period 2π/ω; the exponential factor modulates this parameterization
with either a continually increasing (µ > 0) or continually decreasing (µ <
0) factor.
When µ > 0, the solutions spiral away from the origin as t → ∞ and in
towards the origin as t → −∞. The origin is called an unstable spiral.
When µ < 0, the solutions spiral away from the origin as t → −∞ and in
towards the origin as t → ∞; the origin is then called a stable spiral.
Example 34.5. The system
}
x ′ = −x + 2y
y ′ = −2x − 3y
(34.75)
has trace T = −4 and determinant ∆ = 7; hence the eigenvalues are
λ = 1 2
[
T ± √ ]
T 2 − 4∆ = −2 ± i √ 3 (34.76)