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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2014 • Question 1

Question

Math 6410 • ODE Fall 2014 • Question 1

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

Raw LaTeX

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

Topic Tags

Existence and Uniqueness Theorems