This syllabus is for
Math 316, Fall 2010, taught by Gavin
LaRose. Homework assignments are
also available.
Please Note: the schedule and assignments listed here
are subject to change. They should not change within a week of due
dates, but may change further in advance.
| Mon | Tue | Wed | Thu | Fri |
|---|---|---|---|---|
| Sep 8: Intro., 1.1-1.3 | Sep 9 |
Sep 10:
Read 1.1-1.3, 2.1
(optional: Read 1.4)
Note:
S1.1: meaning of differential equation,
equilibrium solution, and how direction
fields are constructed; S1.2: why a DE has a family
of solutions, meaning of initial condition and
general solution; S1.3: meaning of
ordinary
and partial differential equation, order,
linear and nonlinear differential equation,
and solution of a DE; S2.1: how an
integrating factor allows solution of a (linear,
first order) DE.
|
||
| Sep 13:
Read 2.2-2.3,2.4
(Note: 2.4 also appears 9/17)
Note:
S2.2: which eqns are separable,
integral curves of a differential eqn, the interval
of validity of solns (ex.2,3); S2.3: the derivation
of the "rate out" in ex.1,3; S2.4: theorems 2.4.1, 2.4.2,
why 2.4.2 does not apply in ex.3, general solutions &
nonlinearity.
|
Sep 14 ---------------- |
Sep 15:
Read 2.5
HW 1 Due
Note:
S2.5: meaning of autonomous, and of
equilibrium solutions and critical
points, how to draw a phase line, how this
lets us sketch solutions, meaning of asymptotic
stability and unstable solutions.
|
Sep 16 ---------------- | Sep 17:
Read (2.4,) 2.8
Lab 1
Note:
S2.4: theorems 2.4.1, 2.4.2,
why 2.4.2 does not apply in ex.3, general solutions &
nonlinearity; S2.8: how the integral
equation (3) is related to (2), what the iteration
method is, the four questions we have to answer to
establish the theorem, how ex.1 derives the iterates
& soln to the equation.
|
| Sep 20:
Read 3.1-3.2
Note:
S3.1: form of a linear 2nd order eqn, meaning of
homogeneous, how to solve a linear constant-coefficient
eqn; S3.2: what a differential operator
is, theorem 3.2.1, the three things it says, what
superposition is, definition of the
Wronskian, how it guarantees a general solution
to a linear 2nd order eqn.
|
Sep 21 | Sep 22:
Read 3.3
Lab 1 Due
Note:
S3.3: what Euler's formula is, how to
find real-valued solutions when roots to the
characteristic eqn are complex.
|
Sep 23 | Sep 24:
Read 3.4
HW 2 Due
Note:
S3.4: the difficulty that arises when a characteristic
eqn has repeated roots, the generalization used in ex.1 to
find another solution, what reduction of order is.
|
| Sep 27:
Read 3.5
Note:
S3.5: why Y1(t) -
Y2(t) is a homogeneous solution,
why the general nonhomogeneous solution is
c1 y1(t) +
c2 y2(t) +
Y(t), how the Method of Undetermined
Coefficients proceeds, what we guess for
Y(t) if
g(t) = eat,
sin(bt), cos(bt), or a polynomial, the
summary on p.180 and table 3.5.1.
|
Sep 28 | Sep 29: Lab 2 | Sep 30 | Oct 1:
Read 3.6
Note:
S3.6: the basic idea of variation of
parameters, why in ex.1 we are able to impose the condition
u1'(t) cos(2t) +
u2'(t) sin(2t) = 0, the
derivation of equations (21) and (26).
|
| Oct 4:
Read 4.1-4.2
HW 3 Due
Note:
S4.1: how many initial conditions are needed for an
nth order equation, meaning of fundamental set
of solutions, linear dependence and
independence, example 2, the general solution to a
nonhomogeneous equation; S4.2: how to solve the constant
coefficient nth-order linear homogenous problem, how
to find the nth roots of -1.
|
Oct 5 | Oct 6:
Read 4.3
Lab 2 Due
Note:
S4.3: how MUC is different for higher-order
equations (as compared to 2nd-order), why the correct guess
in example 1 is yp =
t3et, similarly, how
the guesses are obtained for examples 2 & 3.
|
Oct 7 | Oct 8:
Read 5.1,5.2
Proj 1 Due
Note:
S5.1: how a power series solution is similar to the
MUC, the ten results about power series, how to shift the
index of summation in a power series; S5.2: meaning
of an ordinary and singular point, the
form of the series solution at an ordinary point, how to
find the coefficients of the series solution, what a
recurrence relation is, how we must write the
coefficients of P(x), Q(x) and
R(x) to find a series solution around
x0, how many terms we have to find in the
series solution.
|
| Oct 11:
Read 5.3 HW 4 Due
Note:
S5.3: what statement we are justifying in this
section, what we need to be true about p and
q for this to work, meaning of ordinary and
singular point in this context, what the radius of
convergence of a series solution at an ordinary point will
be, the radius of convergence of Q/P.
|
Oct 12 | Oct 13:
Read 5.4
Note:
S5.4: what an Euler equation is, how
solutions are obtained, how solutions are obtained for
repeated and complex roots, why we are unable to just
ignore singular points, what a regular singular
point is.
|
Oct 14 | Oct 15:
Read 5.5
Note:
S5.5: how we rewrite equation (1) to get equations
(3) and (4), the form of the series solution we look for in
the case of a regular singular point, how in example 1 a
solution is obtained from this form.
|
| Oct 18: Fall Break (no class) |
Oct 19: Fall Break (no class) |
Oct 20:
Read 6.1, 6.2 HW 5 Due
Note:
S6.1: what the comparison theorem for improper
integrals says, what an integral transform is,
what the Laplace transform is, the three steps to
using the Laplace transform, how the transforms of
f(t) = 1, eat, and
sin(t) are obtained; S6.2: what the Laplace
transform of f'(t), f''(t), etc.,
are, how we can transform a differential equation and find
its solution by inverting the transform, the advantages of
the transform method, how to use a table of transforms to
solve differential equations.
|
Oct 21 | Oct 22:
Read 6.3, 6.4
Note:
S6.3: what functions appear in some of the more
interesting applications of the Laplace transform, what the
unit step function is, the transform of
uc(t) and
uc(t) f(t-c),
theorem 6.3.2; S6.4: the solution of examples 1 and 2.
|
| Oct 25:
Read 6.5;
Lab 3
HW 6 Due
Note:
S6.5: what an impulse,
I(t) is, and how it is related to the
forcing term g(t) in an equation, how we
use a limiting process to define the unit impulse
function δ, what we get when integrating
δ(t-t0)f(t)
over an interval containing t0.
|
Oct 26 | Oct 27: [Review] | Oct 28 | Oct 29: Midterm Sections 1.1–5.5 |
| Nov 1:
Read 6.6
HW 7 Due
Note:
S6.6:
what L-1{F(s)G(s)
is, the definition of the convolution
(f *g)(t), properties of the convolution
operator, how the convolution theorem is used in examples
1 and 2, what the transfer function for a
differential equation is.
|
Nov 2 | Nov 3:
Read 7.1-7.3
Lab 3 Due
Note:
S7.1: how a second-order equation is transformed
into a system of two first-order equations (ex. 1),
what a solution to a system is, how thm. 7.1.1
is similar to thm. 2.4.2, how thm. 7.1.2 is similar to
thm. 2.4.1; S7.2: how to multiply matrices and
vectors, what invertible and singular matrices are;
S7.3: when solutions to Ax = 0
exist, and how many there are, what eigenvalues
and eigenvectors are, the solutions of example
4.
|
Nov 4 | Nov 5:
Read 7.4
Note:
S7.4: how superposition works for systems, what a
general solution to a system is, a fundamental solution
set, how the Wronskian works with this.
|
| Nov 8:
Read 7.5
Proj 2 Due
Note:
S7.5: what a phase plane and phase
portrait are, how two solutions are found for
example 1, how in example 1 the line x2
= 2 x1 is found, how trajectories on
this line are found, how the phase portrait is then
obtained, what a saddle point and node
are, the general theory for finding solutions using
eigenvalues and eigenvectors.
|
Nov 9 | Nov 10:
Read 7.6
Note:
S7.6: how the real-valued solutions are found from
a complex valued solution in example 1, what a spiral
point is, what cases in solving systems there are
beyond the three on p.404.
|
Nov 11 | Nov 12:
Read 7.7
HW 8 Due
Note:
S7.7: what a fundamental matrix is, what
Ψ(t) (Psi(t)) and Φ(t) (Phi(t)) are and how
they are related, how exp(At) is related to
eat, what an uncoupled system is, how
a matrix A may be diagonalized, how this is used
in example 3 and following to solve x' = A x.
|
| Nov 15:
Read 7.8
Note:
S7.8: what additional solutions to the homogeneous
system looks like when there are repeated eigenvalues
with distinct eigenvectors, and with non-distinct
eigenvectors, what a generalized eigenvector is
and how it is related to this problem.
|
Nov 16 | Nov 17:
Read 7.9
Note:
S7.9: how the transformation x = Ty
simplifies the nonhomogeneous system, when undetermined
coefficients can be used for a nonhomogeneous system, how
this differs from the case of a single equation, how
variation of parameters is more general than
diagonalization or undetermined coefficients, what the
(matrix) equation satisfied by u in variation
parameters is, how this is solved in example 3, how
Laplace transforms are used in example 4.
|
Nov 18 | Nov 19: Review |
| Nov 22:
Read 8.1-8.2
HW 9 Due
Note:
S8.1: what the Euler method is, how it is
related to a difference quotient, what the backward
Euler formula is, what we mean by
convergence of a numerical method, what
truncation and round-off errors are,
how big the global truncation error is for the Euler
method;
S8.2: what the improved Euler formula is,
how the error in the formula depends on step size, how we
might vary step size in a calculation to improve accuracy.
|
Nov 23 | Nov 25: Thanksgiving | Nov 26: Thanksgiving (no class) |
|
| Nov 29:
Read 9.1
Note:
S9.1: what understanding the methods of chp 9
give of solutions to nonlinear systems, what
equilibrium solutions and critical
points are, what the phase plane and a
phase portrait are, how the phase portraits in
cases 1–5 are obtained, how the conclusions on
p.493 and in the table on p.494 reflect the different
eigenvalues found in cases 1–5.
|
Nov 30 | Dec 1:
Read 9.2
Note:
S9.2: the significance of autonomous
systems for our analysis and trajectories in the phase
plane, the meaning of stable,
unstable, and asymptotically stable
critical points, why critical points are interesting
when the behavior of solutions, what a basin of
attraction and separatrix are, how we
(might) be able to solve to find the equations of
trajectories in the phase plane.
|
Dec 2 | Dec 3:
Read 9.3
Lab 4 Due
Note:
S9.3: why stability is of interest in physical
systems, how small perturbations to A may change
its eigenvalues, what we mean by a locally
linear system, how the system (10) may be
determined to be locally linear, what the linearization
of (10) is, what the Jacobian matrix is, why
the linear and locally linear columns of the table on
p.531 differ as they do.
|
| Dec 6:
Read 9.4
HW 10 Due
Note:
S9.4: how the systems in examples 1 and 2 are
analyzed, what a nullcline is.
|
Dec 7 | Dec 8: Lab 5 | Dec 9 | Dec 10: Review Proj 3 Due |
| Dec 13: Lab 5 Due Last class day |
Dec 14 | Dec 15: Final 1:30–3:30pm |
Dec 16 | Dec 17 |