Quizzes For Exam 1
Quiz 1 (due 28 Jan at 11:59PM) [soln posted 11:08AM 29 Jan]: Consider the differential equation $x'=x^2+2x+a$ with parameter $a$. Find all bifurcation points and describe how the number of equilibria change as $a$ varies.
For Exam 2
Quiz 2 (due 19 Feb) [soln posted 10 March at 11:40AM]: Find $e^{At}$ when $A=\begin{bmatrix} 1&1 \\ -2&-1 \end{bmatrix}$.
Quiz 3 (due 7 Feb) [soln posted 10 March at 11:40AM]: Find an equation for the orbits of the system
$\left\{ \begin{array}{ll}
x'=y-1 \\
y'=-xy
\end{array}\right.$
For Exam 3
Quiz 4 (due 27 March): Find constants $\alpha$ and $\beta$ so that $V(x,y)={\alpha}x^2+{\beta}y^2$ defines a Lyapunov function for the system $\left\{ \begin{array}{ll}
x'=-x-5y \\
y'=3x-y^3
\end{array}\right.$
Quiz 5 (due 11 April): Show that $c(-t)=c(t)$ (see 7 April notes).
Quiz 6: In the proof of Theorem 5.31 (in 16 April notes) we claimed that $p(b)x_0(b)x_0'(b) \leq 0$. Prove that claim (hint: it should resemble the proof that $p(a)x_0(a)x_0'(a) \geq 0$ from those notes).