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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2018 • Question 1

Question

Math 6410 • ODE Fall 2018 • Question 1

Let \(f:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuous function and \(x_0\in \mathbb {R}^d\). Show that there is \(T>0\) such that the initial value problem \[ \begin {aligned} \dot {x}&=f(t,x)\\ x(0)&=x_0 \end {aligned} \] has a solution \(x(t)\in C^1([0,T),\mathbb {R}^n)\). Is \(x(t)\) unique? [Don’t just quote a result. State the theorem and give as complete and detailed a proof as you can.]

Raw LaTeX

Let \(f:[0,\infty)\times\R^d\to\R^d\) be a continuous function and
\(x_0\in\R^d\). Show that there is \(T>0\) such that the initial value problem
\[
  \begin{aligned}
    \dot{x}&=f(t,x)\\
    x(0)&=x_0
  \end{aligned}
\]
has a solution \(x(t)\in C^1([0,T),\R^n)\). Is \(x(t)\) unique? [Don't just quote a
result. State the theorem and give as complete and detailed a proof as you can.]

Topic Tags

Distributions and Weak Solutions Existence and Uniqueness Theorems Gronwall Inequality and Continuous Dependence Poincare Maps and Periodic Solutions