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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Spring 2024 • Question 1

Question

Math 6410 • ODE Spring 2024 • Question 1

Let \(f(t,x):[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuous function and \(x_0\in \mathbb {R}^d\). Suppose that there are constants \(L,M<\infty \) such that

\[ |f(t,y)-f(t,z)|\le L|y-z|\quad \text {and}\quad |f(t,0)|\le M \]

for all \(t\), \(y\), \(z\).

Show that the initial value problem

\[ \dot {x}=f(t,x) \]
\[ x(0)=x_0 \]

has a unique solution \(x(t)\in C^1([0,\infty ),\mathbb {R}^d)\). [Don’t just quote a result. State the theorem and give as complete and detailed a proof as you can.]

Raw LaTeX

Let \(f(t,x):[0,\infty)\times\mathbb{R}^d\to\mathbb{R}^d\) be a continuous
function and \(x_0\in\mathbb{R}^d\). Suppose that there are constants
\(L,M<\infty\) such that

\[
|f(t,y)-f(t,z)|\le L|y-z|\quad\text{and}\quad |f(t,0)|\le M
\]
for all \(t\), \(y\), \(z\).

Show that the initial value problem

\[
\dot{x}=f(t,x)
\]

\[
x(0)=x_0
\]


has a unique solution \(x(t)\in C^1([0,\infty),\mathbb{R}^d)\). [Don't just quote a result. State the theorem
and give as complete and detailed a proof as you can.]

Topic Tags

Existence and Uniqueness Theorems Gronwall Inequality and Continuous Dependence Poincare Maps and Periodic Solutions Variation of Constants Formula