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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Spring 2025 • Question 3

Question

Math 6410 • ODE Spring 2025 • Question 3

Let \(f(t,x):\mathbb {R}\times \mathbb {R}^n\to \mathbb {R}^n\) be continuous such that for some

\[ L\in \mathbb {R} \]
\[ |f(t,x)-f(t,y)|\le L|x-y| \]

for all \(t\in \mathbb {R}\) and \(x,y\in \mathbb {R}^n\). Show that the initial value problem for \(x_0\in \mathbb {R}^n\) \[ \dot {x}=f(t,x)\quad \text {with }x(0)=x_0 \] has a unique solution that exists on \(t\in [0,\infty )\).

Raw LaTeX

Let \(f(t,x):\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n\) be continuous such that for some

\[
L\in\mathbb{R}
\]

\[
|f(t,x)-f(t,y)|\le L|x-y|
\]

for all \(t\in\mathbb{R}\) and \(x,y\in\mathbb{R}^n\). Show that the initial value problem for \(x_0\in\mathbb{R}^n\)
\[
\dot{x}=f(t,x)\quad\text{with }x(0)=x_0
\]
has a unique solution that exists on \(t\in[0,\infty)\).

Topic Tags

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