i
l
\
\
x
i the system to loose 80 % of its initial energy? To solve, nd the solution x
Math 2280-001
Week 7 concepts and homework, due Feb 27
Recall that all listed problems are good for seeing if you can work with the underlying concepts; that the
under
Math 2280-001
Fri Feb 13
3.3-3.4.
This week's homework will be due Wednesday at 5:00 pm. and will be drawn from sections 3.3-3.4. Our
rst exam is next Friday February 20, and will cover through section 3.4.
FirstI Leftovers from section 5.3:
Consider the
Math 228001
Wed Jan 28
Q: Autonomous Differential Equations.
Recall, that a general rst order DE for x = x(l ) is written in standard form as
x em x) a
which is shorthand for x (1') =f(t, x(t) ).
Defmition: If the slope function f only depends on the valu
Week 4.3
a)
:> Digits I: 16 :
> evalf(1n(2) );
_ 0.6931471805599453 (1)
:> restart : # clear any memory'om earlier work
:> Digits = 15 : # we need lots of digits for famous numbers
=>
I_> anassign( x, y); # in case you used the letters elsewhere, and c
Math 2280-001
Fri Feb 27
Section 3.6: forced osculations in mechanical
circuits) overview:
We study solutions x(r) to
é'y's'tems (and as we shall see in section 3.7, also in electrical
mx+cx +kx=FO cos(0)r)
using section 3.5 undetermined coefcients algori
Math 2280-00]
Wed Feb 11
3.2-3.3
- Review Monda '5 notes about the eneral theory for nm order linear differential e uations, section 3.2.
3 g q
We talked about the case n r 2 carefully on Monday, but skipped over the generalization details in order to
wor
Math 2280-001
Week 9 concepts and homework Due Friday March 13
sections 4.1, 5.1-5.3
4. I .' modeling coupled mass-Spring systems or main-component inputoutput systems with systems of
dierentiai equationsmonverting single dereniial equations or systems of
P( )d
- Remark: If we abbreviate the function J x x by renaming it G (x), then the formula for the solution
y(x) to the rst order DE above is
y(x)= 62x) lelPWQomH GE)
If x0 is a point in any interval 1 for which the functions P(x), Q (x) are continuou
11 Consider the following initial value problem, which could arise from Newton's second law in a forced
mass-spring oscillation problem:
x"(t) + 2x(t) +1 x(r) =26 sin(2t)
x(0) = O
x (0) I 6.
1_a)_ As a step in solving this IVP, nd the general solution t
Math 2280-001
Mon Apr 20
The idea of Fourier series i
s related to the linear algebra concepts
We'll review this connectio
of dot product, norm, and projection.
n after the denition of Fourier series:
Letf: [-TC, 1r.]>IR bea
2 1c - periodic function.
Exam
Math 2280-001
Mon Feb 23
Section 3.5: Finding y)D for non-homogeneous linear differential equations
My) :f
(so that you can use the general solution y = yP + yH to solve initial value problems, and because
sometimes a good choice for y]D contains the most
Math 2280-00]
Fri Mar 6
4.1, 5.1-5.2 Linear systems of differential equations
- Finish Wednesday's notes to understand why any differential equation or system of differential
equations can be converted into an equivalent (larger) system to a first order d
i
l
\
\
x
i the system to loose 80 % of its initial energy? To solve, nd the solution x
Math 2280-001
Week 7 concepts and homework, due Feb 27
Recall that all listed problems are good for seeing if you can work with the underlying concepts; that the
under
Math 2280-001
Fri Mar l3
5.3 phase diagrams for two linear systems of first order differential equations
xii") 011912 x1
x2([) Q21 22 x2
Our goal is to understand how the (tangent vector eld) phase portraits and solution curve trajectories are
shaped by t
ux/K 4 ' X "\Z X
"X J'LZI, J l X 1/ if Neg/IL .( _
mm < m: u
-I [(7 - .4
Ma):
17(0):»!
5:- V
- 4
>6u : Jik
Ku+qx Pa
1135:
Y=Clw53z fCL5un'3t
* KW): Xo= C.
A X20 0: C =7 CL= 49
C' L 'I A MW! C: 1 U9:
:3 . a: ' o
l + -ml
{03)
{M as an em a Mm w - 'n
x1;
Math 2280001
Mon Mar 9
5.1-5.2
- Finish discussion/examples of the fact that every nth order differential equation or system of differential
equations is actually equivalent to a (possibly quite large) system of first order differential equations This
is
Math 2280-001
Mon Mar 23
- First, do the last exercise in Friday March 13 notes - understanding phase portraits for complex
eigenvalues. Then
5.4 Massspring systems: untethered mass-spring trains, and forced oscillation non-homogeneous
problems.
Consider
Math 2280-001
Wed Mar 2%.?
5.4 Mass-spring systems and forced osciliation non-homogeneous problems.
- Finish Monday's notes, about unforced, undamped oscillations in multi mass-spring congurations. As
a check of your understanding between rst order system
Name . .
ID. number . .
Math 2280-001
Practice Second Midterm
April 1, 2015
This exam is closedbook and closedgncte. You may use a scientic calculator, but not one which is
capable of graphing or of solving differential or linear algebra equations. In ord
Math 22801
Wed January 14 (A)
HW due Friday problem session tomorrow Thursday 8:35-9:25 AM WBB 617 1
Quiz Friday see 2250 page from last spring to see level of questions 7 e f r. \ {4.
U f i .,. \ / i .)<
1.2 Differential equations of the form y (x) :f(x
Math 2280-001
Mon Mar 30
5.5 Linear systems .1 = A )_r for which A is not diagonalizable.
We will spend a large part of the lecture nishing Friday's notes on section 5.5.
Using the Jordan canonical form of a non-diag
Friday's notes, one is led (after omit
Math 2280-001
Numerical Solutions to first order Differential Equations
February 4. 2015
You may wish to bring the .pdfvers ion ofthese notes for class on Wednesday Feb. 4 along to LCB
1 15% where we will work through the Maple version located at
http;,:"
Math 2280-001
Mon Feb 2
2.3 Improved velocig models: velocity-dependent drag forces
For particle motion along a line, with
position x(t) (ory(t) ) ,
velocityx (t) = v(t) , and
accelerationx (t) = v (I) = (1(1)
We have Newton's 2"d law
m v (f) : F
where F
Math 2280-001
Mon Feb 9
- Finish Friday's notes and examples about second order linear differential equations, including Theorems
1,2,3 and the associated examples. The following comments are for us to consider after we finish Exercise
Q there:
Although w
C
Math 2280-001
Quiz 12 I a
April 24, 2015
You may choose. to do #1 (Laplace transform), or #2 (Fourier series). Indicate clearly which problem
you'd like graded.
mm 0 N *3
ind a formula for solutions to the forced oscillation problem below, as convoluti