|
# |
Date Assigned |
Date Due |
Topic |
Hint |
|
Quiz 1 |
Fri
Sep 5 |
Mon Sep 8 |
Introduction to Differential Equations |
Think about the particular solutions in
u variables, what would the initial conditions be then? |
|
Reading Quiz 1 |
Mon
Sep 8 |
Mon Sep 8 |
Sections 1.1-1.4 |
|
|
BONUS Quiz 1 |
Wed
Sep 10 |
Fri
Sep 12 |
Singular Solutions |
What graphical features do you notice
about the singular solution y=0 and the family of solutions y=((x+C)^12)/16
to the ODE y'=x sqrt(y)? |
|
Quiz 2 |
Fri
Sep 12 |
Mon
Sep 15 |
Analyzing The Logistic Equation |
Think about what ideas from Calculus you
can draw upon to sketch a curve y=f(t) accurately. |
|
BONUS Quiz 2 |
Wed
Sep 17 |
Fri
Sep 19 |
Bifurcations |
Recall the quadratic formula for ax^2+bx+c
and how to solve quadratic inequalities, i.e. if |x|<2 that means that -2 <
x < 2. What must be true at a stable equilibrium point of y'=f(y)?
|
|
Reading Quiz 2 |
Mon
Sep 22 |
Mon Sep 22 |
Sections 1.5-1.9 |
|
|
Quiz 3 |
Fri
Sep 26 |
Mon Sep 29 |
Systems
of ODEs |
Use your experience with seeing phase
portraits of Lotka-Volterra in class to help you draw the pictures in the
quiz. Remember properties of logs! log(a)+log(B)=log(AB) and log(e^B)=B. |
|
Reading Quiz 3 |
Mon
Sep 29 |
Mon Sep 29 |
Sections
2.1-2.3 |
|
|
Quiz 4 |
Fri
Oct 10 |
Wed
Oct 15 |
Visualizing 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? |
|
BONUS Quiz 3 |
Fri
Oct 17 |
Mon
Oct 20 |
More Practice with Linear
Systems of ODEs |
Although the question does not explicitly
ask you to, you should find the eigenvalues and eigenvectors of the
associated matrix. You can use their signs to determine the stability of the
origin and thus the trajectories of the given solutions. |
|
Reading Quiz 4 |
Fri
Oct 24 |
Fri
Oct 24 |
Sections 3.1- 3.6 |
|
|
Quiz 5 |
Fri
Oct 24 |
Mon
Oct 27 |
Linear Systems of ODEs and Bifurcation |
HINT: What property of the matrix controls
when a linear system of ODEs will change its character? |
|
Quiz 6 |
Fri
Oct 31 |
Mon
Nov 3 |
Bifurcation in Quasi-Linear Systems |
HINT: Why can't t -> infinity in part (d)?
Your expression for r(t) in both (c) and (d) should have an unknown constant
C in it. |
|
Quiz 7 |
Wed
Nov 12 |
Mon
Nov 17 |
Laplace Transforms |
HINT: 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 in order to obtain
the definition of Gamma[a]. |
|
Reading Quiz
5 |
Fri
Nov 21 |
Fri
Nov 21 |
Sections
6.1-6.5. |
|
|
Quiz 8 |
Mon Nov 24 |
Mon Dec 1 |
Inverse Laplace Transforms |
HINT: Notice the difference in where the
two sums start. Also, note that anything that doesn't have a k in it does not
need to be to the right of the summation symbol. The graphs of b(t) and f(t)=a(t)-b(t)
should be especially pretty. :) |
|
BONUS Quiz 4 |
Mon
Nov 24 |
Mon Dec 1 |
Convolution |
HINT: L(f*f)=F*F=F^2. To find F will
involve taking a square root which leads to the plus/minus symbol. |
