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Phase planes for linear systems with real eigenvalues

Phase planes for linear systems with real eigenvalues. SECTION 3.3. Summary. Suppose Y (t) = e  t V 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 ?

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Phase planes for linear systems with real eigenvalues

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  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)

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