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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Spring 2009 • Question 1

Question

Math 6410 • ODE Spring 2009 • Question 1

Suppose \(f(x,t)\in C^1(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Show that for the initial value problem \begin{equation} \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \tag {1} \end{equation} there exists a unique local solution, i.e., for some \(\epsilon >0\) there is a unique function \(y\in C^1((t_0-\epsilon ,t_0+\epsilon ),\mathbb {R}^n)\) which satisfies (1) for \(t\in (x_0-\epsilon ,x_0+\epsilon )\).

Raw LaTeX

Suppose \(f(x,t)\in C^1(\R^n\times\R,\R^n)\) and
\((x_0,t_0)\in\R^n\times\R\). Show that for the initial value problem
\begin{equation}
\begin{cases}
  \dfrac{dx}{dt}=f(x,t),\\
  x(t_0)=x_0.
\end{cases}
\tag{1}
\end{equation}
there exists a unique local solution, i.e., for some \(\epsilon>0\) there is a unique
function \(y\in C^1((t_0-\epsilon,t_0+\epsilon),\R^n)\) which satisfies (1) for
\(t\in(x_0-\epsilon,x_0+\epsilon)\).

Topic Tags

Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Existence and Uniqueness Theorems Poincare Maps and Periodic Solutions