# Math 316. Applied Ordinary Differential Equations. Fall 2018.

## Books:

Section 002. TR 9:30-10:45. Room: DSH 233. Syllabus for Math 316-002 (18688).
"Differential Equations", by Brannan and Boyce (BB), 2nd edition. "ODEs using Matlab", 3rd ed. by Polking and Arnold (PA). Here you can find John Polking's web-page with the link to Java versions of dfield and other programs.
Here is author's original download page for different MATLAB scripts.
Short MATLAB tutorial by Prof. Lau.

## Office hours:

TR 12:30-13:45. Room: SMLC 220.

## How grades are assigned?

All homeworks: 100 points.
Two midterm exams: 100 points each (100+100 points).
In class quizes: 50 points.
Final exam: 200 points.
Total: 550 points.
Thresholds for grades (not higher than):
A = 495, B = 440, C = 380, D = 330.

Week # Homework problems Due date
11 Final Exam (Tuesday, December 11th, 7:30am).
Quiz 08 (Tuesday, December 4th) (Solution published). You need to be able to find critical points of nonlinear system and analyze them.
Quiz 10 (Thursday, December 6th) (Solution published). Topic: method of undetermined coefficients (Review).
You'll need to be able to propose the right form for particular solution.
Sample final exam problems:
Sec. 2.1(1.2): 18;
Sec. 2.5(2.4): 12 (in addition find implicit analytical solution using partial fractions);
Sec. 3.3(3.3): 13;
Sec. 3.4(3.4): 6;
Sec. 3.5(3.5): 11;
Sec. 4.3(4.3): 41 (reduction of order, not present in 2nd Ed., see lecture);
Sec. 4.4(4.3): 39(59);
Sec. 4.6(4.5): 5, 7, 28;
Sec. 4.8(4.7): 17, 22;
Sec. 5.6(5.6): 6;
Sec. 7.2(7.2): 7.
There will be 8 problems on the final exam. You can have a page (both sides) of information with you.
You shall get Laplace transforms table. No calculators are allowed!
10 Quiz 07 (Tuesday, November 27th) (Solution published). You need to be able to determine type of a critical point of a linear system of ODEs and sketch the phase portrait.
Quiz 09 (Thursday, November 29th) (Solution published). Topic: method of integrating factor (Review).
Sec. 5.7: 1, 2, 5;
Sec. 7.2: 1, 3, 5, 7, 9, 11.
Matlab Problem: Study Example 3 on p.109ff of Polking.
Redo the problem using the initial condition x_1(0)=1, x_2(0)=0.
December 4th, 2018, class time.
10 Second Mid-Term Exam. (Solution published).
You will have a table of Laplace transforms and also you can have ONE page (one side) of whatever you like.

Sample problems for the exam: BB 4.3, p. 252: 55; BB 4.5, p. 273: 7; BB 4.7, p. 292: 16;
BB 5.4: 8, 11; BB 5.6, p. 357: 3. Also the reduction of order technique is recommended for review.

BB 5.2: previously you have solved problems for Section 4.5: 14, 16, 18 on p. 272.
For these problems (from BB 4.5) find the Laplace transform (LT) of the solution as directed on p. 325 for problems 12-21.
Then show it is the same as the LT of the solution determined on p. 273.
BB 5.3: 6, 9-24;
BB 5.4: 2, 4, 6, 8, 10;
(you will need to use the techniques from BB 5.3 to find the inverse transform, as e.g. in problems 9-24 pp. 334-335);
BB 5.5: 7, 9;
BB 5.6: 3, 7, 8.
November 13th, 2018, class time.
9 Quiz 06 (Tuesday, November 1st) (Solution published). You need to be able to use method of variation of parameters for a second order ODE.
BB 4.6: 1, 2, 4, 13, 15, 16, 17, 18;
BB 4.7: 2, 4, 12, 14;
BB 5.1: 4, 16, 20, 24.
November 1st, 2018, class time.
6-8 Quiz 05 (Thursday, October 18th Tuesday, October 23rd) (Solution published). Topic: Homogeneous ODEs of the second order with constant coefficients.
BB 4.1: 1-5 (give reasons);
BB 4.2: 8, 10, 20, 21, 24;
BB 4.3: 1, 2, 14, 4, 41; In 2 and 14(b), sketch the phase plane portrait by hand. You are not asked to use PPlane.
BB 4.3: 6, 12, 23, 42. Note that 42 is an extension of 12;
Extra problem: Explain why the spirals in the context of Sec. 4.4 are always CW;
BB 4.4: 1,2;
BB 4.5: 2, 9, 10, 13, 14, 15, 16, 18, 27;
BB 1.2: 39.
October 18th, 2018, class time.
5 Quiz 04 (Thursday, September 27th) (Solution published). Topics: Systems of linear first order ODEs.
Exam 1 (Solution published)(Example) (Tuesday, October 2nd). This homework is a practice. All previous quizzes will be included as well.
(1) BB 3.4: 2, 7. Include a sketch of the phase plane portrait drawn by hand.
(2) BB 3.4: 10. Include the general solution and a sketch of the phase plane portrait drawn by hand
(3) BB 3.5: 7, 8. Find the general solution and solve the IVP. No plots are required.
October 2nd, 2018, class time.
3-4 Quiz 03 (Thursday, September 13th) (Solution published). Topics: Euler method, local and global errors.
BB 3.1: 14, 16, 18, 22, 34, 38.
BB 3.2 and 3.3: 17 p.150 and 9 p.167
(1) BB 3.2: 22
(2) BB 3.2 and 3.3: 17 p.150 and 9 p.167
(3) BB 3.2 and 3.3: 18 p.150 and 11 p.167 and 16 p.167.
In 16 also find an IC such that x(t) -> 0 as t -> infinity
(4) BB 3.3: 2, 13, 18, 20;
(5) BB 3.4: 3, 8;
In problems (2), (3), (4) and (5) use Polkings PPLANE as discussed in his Chapter 7 (PA)
to draw the direction fields and phase plane portraits.
September 25th, 2018, class time.
2 Quiz 02 (Tuesday, September 4th) (Solution published). Topic: Method of integrating factor.
BB 2.1: 1,9,10,12; use ezplot or fplot for graphs
BB 2.4: 2,4,10,12; In problems 2 and 12 find the solution using the
separable technique and partial fractions and verify your sketch.
BB 2.7 p.109. Use eul.m and rk2.m to check the Table 2.7.1.
rk2.m implements the Improved Euler.
For the following problems (from Polking, p. 72 if you have the book) you have to do two parts:
(a) Find the exact solution (if not mentioned otherwise!);
(b) Use MATLAB to plot on a single figure window the graph of the exact solution(if not said otherwise),
together with the plots of the solutions using each of the three methods (eul.m, rk2.m, and rk4.m)
with the given stepsize h. Use a distinctive marker (type "help plot" for help on available markers)
for each method. Label the graph appropriately and add a legend to the plot.
I. You will need to download rk4 from Polking's web site. Consider the following problem:
x'=x*sin(3t), with x(0)=1, on interval [0,4]; h=0.2.
II. Follow the instructions about MATLAB above (you don't need exact solution!!!) for the IVP
y'=t^2 + y^2, with y(0)=1, on [0,1]; h=.1 and .01. Can you estimate the blow up point?
This problem does not have a solution in terms of elementary functions, it does have a
solution in terms of Bessel functions.
III. Solve the IVPs y'=y^2 and y'=1+y^2 with y(0) =1. Discuss the forward
interval of existence. In this case it corresponds to where the solutions blow up.
Use the direction fields to argue that the solution in II blows up in between the other
two blow up points.
September 11th, 2018, class time.
1 Quiz 01 (Thursday, August 30th) (Solution published). Topics: product rule, chain rule, table integrals, integration by parts.
Download dfield2018a and pplane2018a which you will need later (original (c) by John Polking, adapted for Matlab2018a).
Here is author's original download page for MATLAB scripts.
BB Sec.1.1 (p. 14): 1, 3; use Matlab and dfield2018a to plot direction fields.
Solve each of the following two initial value problems and plot the solutions for several values of y_0.
Then describe in a few words how the solutions resemble and differ from each other:
1a. y'=-y+5, y(0)=y_0;
1b. y'=-2y+5, y(0)=y_0.
previous two problems do in two ways:(I)find solution and plot it using Matlab's ezplot (see p.9 in PA)
(II) use dfield2018a to obtain a plot of the dfield2018a with several solutions
BB 1.2 (p. 25): 13, 15, 16.
BB 1.3 (pp. 36-37): 1, 9, 16; use the eul.m script (see the link for scripts above), do 1 step by hand to check code.
BB 1.4 (p. 42): 8, 13, 14;
For Matlab exercises show your matlab work. BE NEAT for full credit.
August 28th, 2018, class time.