|
#
|
Date
Assigned |
Date
Due |
Topic |
Hint |
Quiz 9
| Mon Nov 10
| Fri Nov 14
| Introduction to Laplace Transforms
| HINT
#1: You basically need to do one step of integration by parts in order
to get the formula for L{t^a}. Think about which function in the
integral you want to differentiate in order to have t^(a-1) appear.
HINT #2: Try integration by substitution (i.e. u=st) in order to obtain the definition of Gamma[a]
|
Quiz 8
| Mon Nov 3
| Wed Nov 5
| Bifurcations in Quasi-Linear Systems of ODEs | HINT:
Notice that you don't have t -> infinity in part (d). Your
expression for r(t) in both (c) and (d) should have an unknown constant
C in it.
NOTE: the contradiction between part (a) saying the system has
acenter for all values of a when in actuality the nature of the fixed
point is very different for different values of a! (i.e. Linearization
doesn't work well when the Jacobian has fully complex eigenvalues or a
zero eigenvalue)
|
BONUS Quiz 3
|
Mon Oct 27
|
Fri Oct 31
|
Visualizing Solutions of Linear Systems of ODEs |
HINT:
Also draw in the straight-line solutions. What does existence and
uniqueness theorem tell you about crossing these solutions? Our general
solution previously discussed applies to zero eigenvalues. What is
e^0?. What will a zero eigenvalue due to the solution trajectories?
|
Quiz 7
|
Mon Oct 27
|
Fri Oct 31
|
Bifurcations in Linear Systems of ODEs |
HINT: What property of the matrix controls when a linear system of ODEs will change its character?
|
Quiz 6
|
Fri Oct 17
|
Wed Oct 22 |
Linear Systems of ODEs |
HINT:
Recall how you can check whether an eigenvector is associated with an
eigenvalue is if it solves Ax=qx where q is an eigenvalue and x is
eigenvector.
|
BONUS
Quiz 2
|
Fri
Sep 26 |
Mon Sep 29 |
Review of Chapter 1 Topics
|
HINT: This is intended to be a quick (15-minute short-answer) quiz.
|
Quiz 5
|
Fri
Sep 26
|
Mon Sep 29
|
Solutions of Linear Differential Equations
|
HINT: Closed form means that you don't have an integral in the expression.
Make sure you fill out the Reality Check to get 1 point! |
| Quiz 4 |
Fri
Sep 19 |
Mon
Sep 22
|
Bifurcations Practice |
When
sketching the bifurcation diagram, think about whether there is
any value of α which would correspond to zero as an equilibrium
value for y*. (That tells you were your bifurcation curve crosses
the horizontal axis in the αy*-plane.) |
Quiz 3
|
Fri
Sep 12
|
Mon
Sep 15
|
Existence and Uniqueness
Theorem
|
Recall that
the domain of definition must be a CONTINUOUS interval of the real line
that include the location of the initial condition.
|
|
|
Fri
Sep 5 |
Mon
Sep 8
|
The Logistic Model of Population
|
The whole
point is that you can sketch a solution of the ODE (representing a
future under the logistic model) for a population WITHOUT solving
for an explicit solution. Remember your rules for curve sketching from
Calc 1!
|
|
Quiz 1
|
Fri Aug 29
|
Wed
Sep 3
|
Introduction to
Differential Equations |
Think
about a second order differential equation and how many unknown
constants its solution would have. What does that imply about its
family of solutions? |
|
BONUS
Quiz 1
|
Fri Aug 29
|
Wed
Sep 3
|
Singular
Solutions |
A singular solution to a DE is another
solution that does not fit into the family of solutions generated by
changing the constant of integration. |
