EDB — 1T3

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Exercises

  1. [1T3] Prerequisites:[1T1],36.Note:Abel’s identity.

    Let be given \(C∈ ℂ^{n× n}\), \(A:ℝ→ℂ^{n× n}\) continuous, and the solution \(Y(t)\) of the ODE

    \[ \frac{d\hskip5.5pt}{d{t}} Y(t) = A(t) Y(t)~ ~ ~ ,~ ~ ~ Y(0)=C \]

    (which has been studied in [1T1]). Set \(a(t)={\operatorname {tr}}(A(t))\), show that

    \[ \det (Y(t)) = \det (C) e^{∫_ 0^ t a(𝜏)\, {\mathbb {d}}𝜏 } \quad . \]

    If \(C\) is invertible, it follows that \(Y(t)\) is always invertible.

    Solution 1

    [1T4]

    [ [1T5]]

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  • Abel
  • Abel identity
  • ODE
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