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Analysis of Linear Systems with Real Eigenvalues: Behavior of Solutions Over Time

This summary explores the behavior of a solution ( Y(t) = e^{lambda t} V ) in linear systems of differential equations as ( t ) approaches infinity. It discusses the effects of the sign of ( lambda ) on the stability of the system, highlighting scenarios where both eigenvalues are negative (leading to a sink) or positive (leading to a source). The case of mixed eigenvalues results in saddle behavior, complicating the dynamics of the system. Exercises provided are useful for reinforcing these concepts using DiffEq software.

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Analysis of Linear Systems with Real Eigenvalues: Behavior of Solutions Over Time

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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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