# Math 316. Applied Ordinary Differential Equations. Spring 2014.

## Books:

Section 005. MWF 09:00 am - 09:50 am. Room: MITCH 221.

"Differential Equations", by Brannan and Boyce (BB) 2nd edition
(1st edition)aaaaaaaa(2nd edition)

aa
"ODEs using Matlab", 3rd ed. by Polking and Arnold (PA).

## Office hours:

MWF 10:00 am - 11:50 am. Room: SMLC 325.

## 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.
Highest thresholds for grades (not higher than):
A = 495, B = 440, C = 380, D = 330.

## Example 2: BB 2.1(1.2): 13,15,16 means that if you have 1st edition you need to do BB 2.1: 13,15,16 , if 2nd -- BB 1.2: 13,15,16

 HW # Homework problems Due date Final Exam (Wednesday, May 14th, 7:30am) There will be 8 problems on the final exam. You can have a table of Laplace transforms and have ONE page (one side) of whatever you like. Quiz 11 (Friday, May 2nd) Topics: 1st order ODEs and systems of two 1st order ODEs Quiz 12 (Friday, May 9th) Topics: 2nd order ODEs (including method of undetermined coefficients. You need to be able to choose the right form for particular solution.) and Using Laplace transform to solve ODEs Sec. 2.1(1.2): 18; Sec. 2.5(2.4): 12 (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; 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. 7 Quiz 10 (Friday, April 25th). You need to be able to find critical points of nonlinear system and analyze them (linearizing nonlinear system around the critical point and determining type of the critical point of the linear system). 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. April 28, 2013, class time. 6 Exam 2 (Monday, April 14th). Included are all topics that we covered in BB 4-5 (Second Order ODEs and the Laplace transform). You can have a table of Laplace transforms and have ONE page (one side) of whatever you like. Sample problems for the exam: Sec. 4.4(4.3), p. 252(252): 35(55); Sec. 4.6(4.5), p. 271(273): 7(7); Sec. 4.8(4.7), p. 290(292): 16(16); Sec. 5.4(5.4): 8(8), 11(11); Sec. 5.6(5.6), p. 354(357): 3(3). Quiz 09 (Friday, April 11th). Topics: Using Laplace transform to solve ODEs Sec.5.2(5.2): previously you have solved problems for Section 4.6(4.5): 16,18 on p.270(272). For these problems (from BB 4.6) find the Laplace transform (LT) of the solution as directed on p.322(325) for problems 12-21. Then show it is the same as the LT of the solution determined on p. 270(272) (which you have got in your previous HW). BB 5.3(5.3): 6, 9-24; BB 5.4(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 p.332(334-335)); BB 5.5(5.5): 7(7), 9(9); BB 5.6(5.6): 3(3), 7(7), 8(8). April 14, 2013, class time. 5 Quiz 08 (Friday, April 4th) Topics: The Laplace transform and its properties.   BB 4.7(4.6): 1,2,4,13,16,17,18; BB 4.8(4.7): 2, 4, 10, 12, 14. BB 5.1(5.1): 4,16,20,24; April 4, 2013, class time. 4 Quiz 05 (Friday, March 7th) Topics: Wronskian. Quiz 06 (Friday, March 14th) Topics: Homogeneous ODEs of the second order with constant coefficients. Quiz 07 (Friday, March 28th) Topics: Nonhomogeneous ODEs of the second order and Resonance phenomenon. (4.5-4.6) BB 4.1(4.1): 1-5 (give reasons); BB 4.2(4.2): 8, 10, 20, 21, 24; BB 4.3(4.3): 1, 2, 6(14), 10(4), 28(41); Supplementary material: In 2 and 6(14)b, sketch the Phase Portrait by hand. You are not asked to use PPlane. In 10b use PPlane. BB 4.4(4.3): 12(6), 14(12), 17(23), 26(42). Note that 26(42) is an extension of 14(12); Extra problem: Explain why the spirals in the context of Sec. 4.4 always go clockwise; BB 4.5(4.4): 1,2; BB 4.6(4.5): 2, 9, 10, 13, 14, 15, 16, 18, 27; BB 2.1(1.2): 39. March 24th, 2014, class time. 3 Quiz 04 ( Friday, February 21th) Topics: Systems of two 1st order ODEs. Exam 1 (Monday,February 24th). Included are all topics that we covered up to now (up to BB 3.5). (1) BB 3.4(3.4): 2,7. Include a sketch of the phase plane portrait drawn by hand. (2) BB 3.4(3.4): 10. Include the general solution and a sketch of the phase plane portrait drawn by hand (3) BB 3.5(3.5): 7,8. Find the general solution and solve the IVP. No plots are required. February 24th, 2014, class time. 2 Quiz 02 ( Friday, February 7th) . Topics: Method of integrating factor. Quiz 03 ( Friday, February 14th) . Topics: Euler method, local and global errors. Eigenvalues and eigenvectors of a 2x2 matrix.   BB 2.7 p.109. Use eul.m and rk2.m to check the table on 2.7.1. (rk2.m implements the Improved Euler) For the following problems from Polking 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. Problem 1 p.72 Polking. (You will need to download rk4 from Polking's web site) 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. 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. IV. BB 3.1: 14,16,18,22,34,38. BB 3.2 and 3.3(3.3): 17 p.150 and 9 p.162 (167) (1) BB 3.2: 22 (2) BB 3.2 and 3.3(3.3): 11 (17) p.147(150) and 9 p.162 (167) (3) BB 3.2 and 3.3: 12 (18) p.147(150) and 11 p.162 (167) and 16 p.162 (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. February 21th, 2014, class time. 1 Quiz 01 (Friday, Jan 31). Topics: product rule, chain rule, table integrals, integration by parts. Download any version of dfield you like (dfield8 for example) and any version of pplane (pplane7 for example) from Polkings web site (see PA last page of preface and p.47) or use http://math.rice.edu/~dfield/ BB Sec.1.1 (1.1): 1, 3; use Matlab and dfield to plot dfields (see Chapter 3in PA) 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 Polkings dfield to obtain a plot of the dfield with several solutions BB 1.3(1.3): 1, 9, 16; use the eul.m script on page 73 (or get it from http://math.rice.edu/~dfield/), do 1 step by hand to check code. BB 1.4(1.4): 8, 13, 14; BB 2.1(1.2): 13, 15, 16. For Matlab exercises show your matlab work. BE NEAT for full credit. BB 2.2(2.1): 1,9,10,12; use ezplot or fplot for graphs BB 2.5(2.4): 2,4,10,12; In problems 2 and 10 find the solution using the separable technique and partial fractions as illustrated on p.86 (82) and verify your sketch. Feb 3, 2014, class time.