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: 1^{st} order ODEs and systems of two 1^{st} order ODEs
Quiz 12 (Friday, May
9th) Topics: 2^{nd} 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 45 (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
1221.
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, 924;
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 924 p.332(334335));
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. (4.54.6)
BB
4.1(4.1): 15 (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 1^{st} 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.

