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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Spring 2025 • Question 2

Question

Math 6410 • ODE Spring 2025 • Question 2

Consider the family of ODE systems

\[ \dot {x}_\epsilon =f(t,x_\epsilon )+\epsilon g(t,x_\epsilon ) \]

with \(\|Df\|\le L\) and \(|g|\le M\). Show that

\[ |x_\epsilon (t)-x_0(t)| \le e^{Lt}|x_\epsilon (0)-x_0(0)|+\frac {M\epsilon }{L}e^{Lt}. \]

Hint: Look at the integral equation for \(z_\epsilon (t):=x_\epsilon (t)-x_0(t)\), estimate using Grönwall-type ideas.

Raw LaTeX

Consider the family of ODE systems

\[
\dot{x}_\epsilon=f(t,x_\epsilon)+\epsilon g(t,x_\epsilon)
\]

with \(\|Df\|\le L\) and \(|g|\le M\). Show that

\[
|x_\epsilon(t)-x_0(t)|
\le e^{Lt}|x_\epsilon(0)-x_0(0)|+\frac{M\epsilon}{L}e^{Lt}.
\]

Hint: Look at the integral equation for
\(z_\epsilon(t):=x_\epsilon(t)-x_0(t)\), estimate using
Gr\"onwall-type ideas.

Topic Tags

Gronwall Inequality and Continuous Dependence