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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1996 • Question 12

Question

Math 6410 • ODE Fall 1996 • Question 12

Apply this theorem to the system \[ \begin {aligned} u'&=-v\\ v'&=-v+g(u)\\ x&=\begin {pmatrix}u\\v\end {pmatrix}, \end {aligned} \] where \(g:\mathbb {R}\to \mathbb {R}\) is a \(C^1\) function such that \[ ug(u)>0,\quad u\ne 0, \] using \[ \phi (u,v)=\frac {v^2}{2}+\int _0^u g(\xi )\,d\xi . \] What is your conclusion?

Raw LaTeX

Apply this theorem to the system
\[
\begin{aligned}
u'&=-v\\
v'&=-v+g(u)\\
x&=\begin{pmatrix}u\\v\end{pmatrix},
\end{aligned}
\]
where \(g:\R\to\R\) is a \(C^1\) function such that
\[
ug(u)>0,\quad u\ne0,
\]
using
\[
\phi(u,v)=\frac{v^2}{2}+\int_0^u g(\xi)\,d\xi.
\]
What is your conclusion?

Topic Tags

Gronwall Inequality and Continuous Dependence