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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Spring 2011 • Question 1

Question

Math 6410 • ODE Spring 2011 • Question 1

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

Raw LaTeX

Let \(f(x,t)\) be continuous for \((x,t)\in\R^n\times\R\) and let \(x_0\in\R^n\) be a
point. Show that there is and \(\epsilon>0\) and a function
\(y\in C^1([0,\epsilon],\R^n)\) 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 Gronwall Inequality and Continuous Dependence Point, Continuous, and Residual Spectrum