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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2010 • Question 1

Question

Math 6410 • ODE Fall 2010 • Question 1

Suppose \(f(x,t)\) is continuous for \((x,t)\in \mathbb {R}^n\times \mathbb {R}\) and let \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\) be a point. Assume that there is a constant \(L<\infty \) so that \[ \lvert f(x_1,t)-f(x_2,t)\rvert \le L\lvert x_1-x_2\rvert \] for all \(x_1,x_2,t\). Show that there is and \(\epsilon >0\) and a unique function \(y\in C^1([t_0,t_0+\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \]

Raw LaTeX

Suppose \(f(x,t)\) is continuous for \((x,t)\in\R^n\times\R\) and let
\((x_0,t_0)\in\R^n\times\R\) be a point. Assume that there is a constant
\(L<\infty\) so that
\[
  \lvert f(x_1,t)-f(x_2,t)\rvert \le L\lvert x_1-x_2\rvert
\]
for all \(x_1,x_2,t\). Show that there is and \(\epsilon>0\) and a unique function
\(y\in C^1([t_0,t_0+\epsilon],\R^n)\) that satisfies the initial value problem
\[
  \begin{cases}
    \dfrac{dx}{dt}=f(x,t),\\
    x(t_0)=x_0.
  \end{cases}
\]

Topic Tags

Existence and Uniqueness Theorems Gronwall Inequality and Continuous Dependence Point, Continuous, and Residual Spectrum Variation of Constants Formula