Mathematics Qualifying Exam Question Bank

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Question

Math 6410 • ODE

Fall 2003 • Question 1

Question

Math 6410 • ODE Fall 2003 • Question 1

Peano Existence Theorem. Suppose \(f(x,t)\in C(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Consider the system of ODE’s \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \qquad (\spadesuit ) \]

Find an example that shows that the solution of \((\spadesuit )\) may not be unique. Give another example that shows that the solution may not exist for all time.
Prove the Peano Existence theorem: Suppose \(f(x,t)\in C(\mathbb {R}^n\times \mathbb {R})\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Then there is an \(\epsilon >0\) and a function \(y\in C^1([t_0,t_0+\epsilon ],\mathbb {R}^n)\) which solves \((\spadesuit )\).

\textbf{Peano Existence Theorem.}
Suppose \(f(x,t)\in C(\R^n\times\R,\R^n)\) and \((x_0,t_0)\in\R^n\times\R\). Consider the
system of ODE's
\[
\begin{cases}
  \dfrac{dx}{dt}=f(x,t),\\
  x(t_0)=x_0.
\end{cases}
\qquad(\spadesuit)
\]
\begin{enumerate}
\item[(a.)] Find an example that shows that the solution of \((\spadesuit)\) may not be
unique. Give another example that shows that the solution may not exist for all time.
\item[(b.)] Prove the Peano Existence theorem: Suppose \(f(x,t)\in C(\R^n\times\R)\) and
\((x_0,t_0)\in\R^n\times\R\). Then there is an \(\epsilon>0\) and a function
\(y\in C^1([t_0,t_0+\epsilon],\R^n)\) which solves \((\spadesuit)\).
\end{enumerate}

Topic Tags

Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Existence and Uniqueness Theorems Poincare Maps and Periodic Solutions