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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2011 • Question 1

Question

Math 6410 • ODE Fall 2011 • Question 1

Let \(f(t,x)\) be continuously differentiable for \((t,x)\in \mathbb {R}\times \mathbb {R}^n\) and let \(x_0\in \mathbb {R}^n\) be a point. Show that there is and \(\epsilon >0\) and a unique function \(y\in C^1([-\epsilon ,\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]

Raw LaTeX

Let \(f(t,x)\) be continuously differentiable for \((t,x)\in\R\times\R^n\) and let
\(x_0\in\R^n\) be a point. Show that there is and \(\epsilon>0\) and a unique function
\(y\in C^1([-\epsilon,\epsilon],\R^n)\) that satisfies the initial value problem
\[
  \begin{cases}
    \dfrac{dx}{dt}=f(x,t),\\
    x(0)=x_0.
  \end{cases}
\]

Topic Tags

Existence and Uniqueness Theorems