NETMaths
Part BCSIR NET June 2023eigenvalue-growth-rates

Eigenvalue growth rates

Suppose x(t) is the solution of the initial value problem in : ẋ = Ax, , where A = [[5, 4], [1, 2]]. Which of the following statements is true?

  1. A.x(t) is a bounded solution for some .
  2. B.|x(t)| → 0 as , for all .
  3. C.|x(t)| as , for all .
  4. D.|x(t)| → 0 as , for all .

Solution

Eigenvalues of A are 6 and 1 (trace 7, det 6), both positive: every non-zero solution grows like or . So (1) fails, (2) fails along the 6-eigendirection, (3) fails along the 1-eigendirection, and (4) holds since 10 > 6.

The trap it tests

Numerical convergence

A scheme assumed to converge, or to converge at its advertised rate.

Drill statements like this

Related counterexample: W(f, g) ≡ 0 implies f and g are linearly dependent

From Ordinary Differential EquationsLinear ODE, Wronskian, variation of parameters, systems

ShareWhatsAppTelegram