- E5
[1QR] Prerequisites:[19S].Let su fix \(x_ 0,t_ 0∈ℝ\), and a bounded and continuous function \(f:ℝ→ℝ\), with \(f(x_ 0)=0\) but \(f(x){\gt}0\) for \(x≠ x_ 0\). We want to study the autonomous problem
\[ \begin{cases} {x}’ (t) = f(x(t))~ ~ , \\ x (t_ 0 ) = x_ 0 ~ ~ .\end{cases} \]Note that \(x≡ x_ 0\) is a possible solution. Show that if, for \(\varepsilon {\gt}0\) small, 1
\begin{eqnarray} ∫_{x_ 0}^{x_ 0 + 𝜀} \! \frac{1}{f (y)}\, {\mathbb {d}}y = ∞ \label{eq:Osgood_ dei_ poveri_ dx} \\ ∫_{x_ 0-\varepsilon }^{x_ 0 } \! \frac{1}{f (y)}\, {\mathbb {d}}y=∞ \label{eq:Osgood_ dei_ poveri_ sx} \end{eqnarray}then \(x≡ x_ 0\) is the only solution; while otherwise there are many class \(C^ 1\) solutions: describe them all.
Solution 1Conditions ?? and ?? are a special case of Osgood uniqueness condition, see Problem 2.25 in [ .
EDB — 1QR
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Authors:
"Mennucci , Andrea C. G."
.
Bibliography
Book index
- [29] Gerald Teschl. Ordinary differential equations and dynamical systems, volume 140. American Mathematical Soc., 2012. ISBN 978-0-8218-8328-0. URL http://www.mat.univie.ac.at/~gerald/ftp/book-ode/index.html. (Freely available on the author’s website).
Book index
- Osgood uniqueness condition
- ODE
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