EDB — 1QZ

Exercises

1. [1QZ] Prerequisites:[1QV]↺↻.Let us fix \(𝛼{\gt}1\), and consider again

   \[ \begin{cases} x’ (t) = |x(t)|^𝛼~ ~ , \\ x (0 ) = 1 ~ ~ \end{cases} \]

   We have seen in [1QV]↺↻ that this ODE admits a maximal solution \(x : I_𝛼 →ℝ\). Fixed \(t∈ℝ\), show that \(t∈ I_𝛼\) for \(𝛼{\gt}1\) close to \(1\), and that \(\lim _{𝛼→ 1+} x(t)=e^ t\).

   Note that \(e^ t\) is the only solution of \(x' (t) = |x(t)|\) with \(x (0 ) = 1\).

   Solution 1

   [1R0]↺↻