EDB — 1T3

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]↺↻]