### Class diary and homework assignments:
M231 MWF 11:15-12:05

**W Aug 24:**
Intro. What *is* an ODE (a PDE)? order; linear/nonlinear
*Hwk 1.1: 1-12 due Fri*

**F Aug 26:**
Solution concept; IVP; radioactive decay; Picard-Lindelöf
*Hwk 1.1: 13,14,15; 1.2: 1,3,5, due Mon *

**M Aug 29:**
Example Picard-Lindelöf & Peano; implicit solution;
direction fields.
*Hwk 1.2: 9,10,15,19,20a,21a,22a,23,25,26,29 and 1.3: 1,3,5,7 due Fri
if possible*

**W Aug 31:**
Euler's approximation method; separable ODEs.
*Hwk: from notes pg3 fill in table. Due Fri. Hwk from Mon until 20a is
definitely due Fri; the rest from Monday may be turned in Wed,
if you prefer. But on Fri, there will come other hwk due Wed.*

**F Sep 02:**
The example y'=1+y^{2}: finite time blow-up; Sol's to IVP's are
defined on *intervals* only. --- First outlook on linear 1st order ODEs.
*Hwk: 2.2: 1-5,7,12,17-19,21; 27b,29,31a-c due Wed. In 17-19,21,
make sure IN ADDITION to what the book asks, to give the *interval* on
which sol's are defined.*

**M Sep 05:** LABOR DAY

**W Sep 07:**
Linear 1st order ODEs: finding and using an integrating factor.
Brief example of a different kind: velocity as an integrating factor in
Newtonian mechanics:
*New Hwk: 2.3: 1-6, 8,11 due Fri. 16-18, 20, 25a, 28 due Mon
*

**F Sep 09:**
Review velocity as int'ing factor in Newtonian mechanics; questions about
hwk; homogeneous ODEs and the v=y/x substitution, one example sketched.

**M Sep 12:**
More substitutions (in particular Bernoulli). Another example like 2.2,27:
How to use *definite* integrals to get the right constant for IVP's when
the integral cannot be evaluated practically. *Hwk: 2.6: 10,13,18,21,22
due Wed --
Review pblms pg81: 1,2,4,8,9,15,17,32. Give interval of existence with 32;
due Fri -- also scan all the review pblms to see in which of the studied
categories (if any) they fit. This latter part is not for turn in *

**W Sep 14:**
Modelling: mixing problems.
*Hwk: 3.2: 1,3,4,5,8 due Mon. --- Extension given for 2.6 hwk till Fri*

**F Sep 16:**
Example for superposition principle in the solution of a mixing
problem's ODE. -- Population models; heating/cooling ---
*Hwk: 3.2: 9,14,19,27 due Mon *

**M Sep 19:**
Q&A; Modelling with Newton's equations if forces depend on velocity
only.
*Hwk: Sec 3.3: 1,3,5,7,12,15. Sec 3.4: 3 --- sol's will be posted but hwk
will not be turned in for grading, so you have the feedback before the
exam*

**W Sep 21:**
Using the Newton method to determine hit-ground time, and when we get
it cheaper. Dimensionless time variable. Quick overview for exam.
The superposition principle for linear ODEs started.

**F Sep 23:** Outline Linear ODEs: splitting
``general solution of lin'inhom = general solution of lin'hom
plus particular sol of lin'inhom''; one example to illustrate this.
*Hwk: You won't turn this in, just do it by next Fri: Evaluate the
complex-number expression* (2+i)/(3-4i) + (2-i)/(3+4i). * (I may have
had different numbers in class, but one is as good as the other.)
Also evaluate
* (cos a + i sin a)(cos b + i sin b),
*using trig identities to simplify
the result. *

**M Sep 26:** ****** EXAM 1 ******

**W Sep 28:** Exam back, and discussion.
Exponential ansatz for lin'hom' ODE with const coeffs.
Why we get *all* solutions from linear combinations of this ansatz.
*Hwk: Ch4.2: 1-4,7,8,13,14, 41,42,48 due Mon. Postpone 3,4 until after
Friday's class, all the rest you can do now. Also remember last Friday's
hwk, and do it by this Friday.*

**F Sep 30:** Linear independence, and
the Wronskian. Double / multiple roots of the characteristic equation
and how the solution arises in the limit of coalescing single roots.

**M Oct 03:** Complex roots of the
characteristic equation, and Euler's formula: *Hwk Ch4.3: 10, 20, 22, 32,
33, 35, 37b. Due Fri (Try some more of the smaller numbers 1-20 without
turn-in, if you need more practice.)*

**W Oct 05:** Examples on complex calculations,
including (for later use) integral of exponential*(sin or cos).
Sneak preview of method of undetermined coefficients

**F Oct 07:** Method of undetermined
coefficients. *Hwk: Ch 4.4: 1-8, 10,11,13,16,17, 29-31,34.
Ch 4.5: 25,28,30 due Wed*

**M Oct 10:** Method of undetermined
coefficients; ramifications. An example on piecewise defined rhs introduced.
*Hwk: Ch.4.5: 32-37, 39, 45 due Wed or Mon*

Link to extra credit problems

**W Oct 12:** Introduction to Variation of
Parameters: comparison with 1st order, and one full-size example.
*Hwk: Ch4.6, 1,6,10,22,24 due Fri next week*

**F Oct 14:** FALL BREAK

**M Oct 17:** Comments on IVP for inhom. ODEs:
determine C1, C2 by plugging initial values into solution of
*in*homogeneous ODE! --- Another example on varition of parameters
(3rd order; from book p 340, but solving for v1, v2, v3 the pedestrian way.
*(remember) Hwk: Ch4.6, 1,6,10,22,24 due Fri*

**W Oct 19:** Writing A cos x + B sin x as
R sin(x+phi). The free oscillator: discussion, calculation, interpretation.

**F Oct 21:** Informal background info
on periodic forcings, which can be written as superpositions of trigs.
Forced oscillator: complex calculation

**M Oct 24:** Forced oscillator:
Interpreatation and Resonance phenomena

**W Oct 26:** Review on Variation of
Parameters; brief introduction to Laplace transform methods (Ch. 7)

**F Oct 28:** EXAM 2

**M Oct 31:** Calculating a LT from scratch.
Linearity properties and LT of a derivative.
*Hwk: Ch 7.2: 2,3,4,6,8,10,11,14,20,31 due Fri*

**W Nov 02:** Exam back; More properties of
the LT. *Hwk Ch 7.3: 2,4,6,8,10,12,14,16,18; 31 due Mon*

**F Nov 04:**Solving a simple IVP via LT.
Can two different fcts have the same LT?
(in principle yes, but for practical purposes: no - not two different
CONTINUOUS fcts!) Is every F(s) somebody's LT? (No way. However those
F(s) we get from attempting to solve IVP's are.) PFD just begun.

**M Nov 07:** PFD: Cover-up method and its
variants in the presence of multiple, or complex roots.

**W Nov 09:** * Hwk 7.4:
2,4,6,8,10,15,16, 21,22,27,28, 33,36 due Fri or Mon.
Hwk 7.5: 3,4,7,8,23,28 due Mon* Solving IVP's via LT: primarily const
coeff's; plus an example where coeff's are no worse than at+b
(leading to a 1st order ODE for Y(s)).

**F Nov 11:**
*More hwk 7.5: 37,38, also due Mon* Q& A; another refinement on PFD
(pg 6,7 item (ii) of PFD notes). The unit step function introduced
(cf p38 of notes)

**M Nov 14:** How to do LT's of piecewise
defined functions and solve IVP's with such fcts on the rhs.
*Hwk 7.6: 1-5,9,12,14,16,18,27,28. This hwk will not be graded, so that
you don't need to relinquish your solutions during exam prep. But
you should do it before the exam, and solutions will be posted
for your comparison. 27 and 28 needs Wed's class yet*

Nov 15: Last day to drop with WP/WF

**W Nov 16:** 7.6 pblm39 finished up.
Review of geometric series. The LT of |sin t| from scratch: how a geom'
series arises in the LT of a period fct from the LT of the
unit step functions. (Brief rmk how they implement |sin t|
voltage in El.Engr)

**F Nov 18:** Q&A -- TCE

**M Nov 21:** EXAM 3

**W Nov 23:**

**F Nov 25:** THANKSGIVING

**M Nov 28:**

**W Nov 30:**

**F Dec 02:**

**M Dec 05:**

**W Dec 07:** STUDY PERIOD

**F Dec 09:** FINAL EXAM 10:15-12:15