### Thoughts, observations, questions

December 19, 2008

I wanted to try to take what I have learned since September – and kind of summarize it.  I wanted also to be able to put some order or a sense of what I learned and what I didn’t understand.  I also wonder how everything fits together with all my previous knowledge – and have to ask if anything in my thinking changed because of this class.

Initially (still a little now) I felt technologically challenged 🙂  But, I also had to realize that I am not fresh out of high school – it was the first time that I had a class were we had to blog.  So, I worked through learning things – I didn’t learn it all at one time – just last week I got bold eneough to experiment with the background and formatting of my blog.  There are a few more things I’d like to learn – but, I am finally okay with not knowing everything right now.  I have learned to keep reminding myself – if I knew everything – there would be no need for school – wow – imagine that 🙂

The first day of class I was still learning short-cuts on the key-board.  Now I do things so quickly and it seems so easy compared to the beginning.  That feels really good.  I remember not having a problem with LaTax in the begining – but, kind of struggling in project four.  Since we couldn’t seem to get things to transfer – I kind of just pasted and cut equations in … not ideal – but, I learned to do what will work and not get stuck.

I found Matlab really difficult for some reason – as I am reading Matlab DeMystified – it seems more logical to me.  I seem to have difficulty just doing something without learning the why’s and how’s – so, as easy as it is to just use a line or two of code – I really tried to learn from my mistakes – from the aspect of – why didn’t it work – what went wrong – what does that error code mean – plus learning the math at the same time – it was very challanging at times – but good.

We also did a bit of Mathematica and TeMath.  I liked the graphical abilities of these programs a lot.

Even though part of me felt like my learning was all over the place – or five things at once – it challenged me and pushed me – so, I ended up at a different level – which I totally appreciated.  I feel like I really matured mathematically through this class.

As I sit and think – I realize the connections from a lot of previous classes and readings.  Euler’s Method – Series – integration and differentiation – all became clearer to me.

At the beginning of the class Dr. Davis talked about learning and thinking about O.D.E.’s analytically and  behaviorally – when you cannot get an exact solution.  We studied linear and non-linear systems and systems of equations.

It was nice to let the computer do the solutions – and learn the theory – the how’s and why’s of using the computer and various software.  Once I got the hang of programming the cells in Excel – it seemed so easy – seems like the first week was a very long time ago.

We also learned some techniques if we could not get exact solutions using the computer.  We learned to look at behavior.  This allowed us to use quantitative techniques – I really enjoyed how Jay compared ode23 with ode45 – it showed visually margins of error – we got to see it numerically in Euler’s Method – the smaller the increments the more accurate.  But we could also see error in the qualitative behavior.  I still want and need to look at and research the Dirac function from the ode that was not separable in project four – I wonder why this particular function was chosen – I learned how to create slope fields – visually put together – solution curves and slopes for me – it added to my repertoire of uses for slope.  Project three helped synthesize theoretically for me how linear algebra techniques are very useful in D.E.  I would have never thought at the beginning – we would end up using matrices and determinants.  It was pretty cool.

I still feel I need to sit with LaPlace transforms for a while.  I don’t feel I’ve done enough of them – because it is not intuitive enough for me.  I am still struggling through them … but, just need to sit and solve a bunch.

As far as analytical techniques – we learned about substitution a and separations of variables.  I enjoyed learning especially in this class – to step away when I am stuck and figure out what I do know first.  Then approaching the problem from there – I guess I feel I matured mathematically most because of this … I now think before I do any solving – I used to just jump in and write .  Now I recognize that – I can ask myself – what does this problem remind me of?  Have I ever seen anything like it?  How do I want to start?  Finish?  I guess I kind of find a plan of attack now – a set of blue prints if you will.  And if the blue prints aren’t working they can be redesigned  and it’s not the end of the world.  I will always see Matt in a red convertible saying “friggin” 🙂

There are still some questions about things like the predator prey model – how do we calculate the unpredictable – like a fire etc.?  Seems like we would have to take so many variables into consideration – that we would be likely to miss something.  The model seems to revolve around just the numbers and animals that exist – not why or how.  I imagine it would be a huge problem if we did this – so, I guess I need to research this more.  We saw Lorenze attractors and Rossler’s.  I am trying to figure out if a predator prey model could ever be 3d – I am assuming this is true by what I know so far – but, if we have four linked d.e. does that put us into 3D?  Going by Linear Algebra so far – 4 eq. with 4 unknowns would say yes – is that all there is to it?  I wonder how over determined and under-determined equations fit in here?  By adding more variables to a predator prey system – do we ever get chaos?  hmmmm – is that were overdet. and underdet. fit in?  Can computers model past 3D?  We are looking at a 2D screen seeing 3D – do we need a 3D screen to see 4D?  Lap-tops would then be lap-boxes – I imagine that is what holograms are – probably my fascination behind changing the axis’ and getting different views of the graph –

### revisit project 2

December 11, 2008

### Finnegan Begin Again

December 9, 2008

Project Four   December 9, 2008

I did a fair amount of reading to be able to better understand LaPlace Transforms to get an idea of  why and when they are used.  The formulas looked rather daunting at first, but as I read through a couple of sections in different books (the two texts recommended for the class and Schaum’s Outlines, and the web page S.O.S. Math – I got a better sense of what was happening theoretically not only with using LaPlace Transforms, but with varies terms and definitions – some that I have worked with but, some others  were new to me.  Since Matt ( my project partner ) so graciously and quickly caught me up to speed on what I had missed  – I felt like I needed to kick into gear very quickly to be a better support to him.  It’s nicer to keep things in balance.  I read through the class info. that was provided to us on project four and the project guide #4.

I began by writing down words and phrases that I didn’t fully understand.   I also wanted to make sure what I thought I knew – I really did know.  I began asking myself questions like – why were we given these equations rather than some others – what was their significance to differential equations – what was important about them?  What would solving them reveal?  Where there any similarities between them?  Or did each equation represent or teach us something different?  Then as read, I realized that I could also compare these equations to those in previous projects – to talk about similarities and differences not only in the equations themselves – but, the methods used to solve them – why one method was better to use than another – things to look for so that I can eliminate methods that won’t work for example LaPlace Transformations would be used for linear differential equations that are exponentially bounded.  The difference between an explicit solution and a general solution – which brought me back to describing things three ways – analytically, qualitatively and numerically.  I had asked myself during the first lecture for this project – are there other uses for LaPlace transforms?

As I read – I began to see some connections between Differential Equations, Linear Algebra and LaPlace Transforms – I read about different functions that I had heard about – but never really quite understood really – I began to question what kind of functions we had – just a brief list (not conclusive) functions like the Gamma function,  Beta function, Dirac function, Sterling’s formula … etc.  As I read about phase planes and damped and undamped harmonic oscillators – then I saw spiral sinks, straight line solutions and equilibrium points – everything that seemed so disjuncted started to not seem so jumbled and disconnencted.  I might have to draw a mapping of how it all connects to see it visually in order to communicate it effectively to Matt so that we can discuss each equation in class.  I am hoping also to be able to answer the extra credit portion – LaPlace Transforms can be used to solve systems of linear differential equations – sets of two or more differential equations – with an equal number of equations and unknowns.  For each differential equation in the system – the LaPlace transform of the unknown functions is taken – you get a set of equations – then the inverse transform is calculated – and you get the explicit answer.  The solution seems to use Linear Algebra with the LaPlace Transform … so, a lot to discuss with Matt  – better get drawing and organize my thoughts for class. TTFN

### Project three graphs

November 6, 2008

Today we were able to get graphs of the different roots – Dr. Davis showed us a technique in Mathematica that let the computer generate diffeent matrices for us. It saved a lot of time because we had discussed just playing with matrices to get the desired results. It was much faster and more efficient to use Mathematica. We had gotten stuck trying to graph the second to last eigenvectors, Dr. Davis looked at the code and was able to see our error and explained initializing cells which was also very enlightening and useful for future endeavors.

% this is how to find the exact solutions –> dsolve(‘Dy= the equation’, ‘t’)

November 4, 2008

### 3rd Project – Nov.4th

November 4, 2008

Today – spent time understanding how to get eigenvectors and eigenvalues – looking up definitions to understand the behaviors of spiral sources and equilibrium points – found a nice picture to compliment our picture

### Hmmmmm soon to revise

October 21, 2008

—————————————————————————————–

99/30/08
9/30/08)

>> y_int=0.1;
>> [t,y]=euler(‘test_example’,[0.1,9.0],y_int,200);
>> y_init=[.9,1,1.1];
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,1000);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,2500);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,3000);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,500);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,800);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,700);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,750);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,799);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,800);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,850);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,830);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,810);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,805);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,1500);
>> plot(t,y)
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,1500);
>> [t,y]=euler_system(‘lorenz_system’,[0.0,20.0],y_init,1000);
>>————————————————————————————

we created m files

g.m

function xdot = g(t,x)

xdot = zeros(3,1);

sig = 10.0;

rho = 28.0;

bet = 8.0/3.0;

xdot(1) = sig*(x(2)-x(1));

xdot(2) = rho*x(1)-x(2)-x(1)*x(3);

xdot(3) = x(1)*x(2)-bet*x(3);

created lorenze_demo.m

function lorenz_demo(time)

% Usage: lorenz_demo(time)

% time=end point of time interval

% This function integrates the lorenz attractor

% from t=0 to t=time

[t,x] = ode45(‘g’,[0 time],[1;2;3]);

disp(‘press any key to continue…’)

pause

plot3(x(:,1),x(:,2),x(:,3))

print -deps lorenz.eps

code for the lorenz attractor animated

function yprime = lorenz_system ( t, y )

function createfigure(XData1, YData1, ZData1, XData2, YData2, ZData2, XData3, YData3, ZData3)
%CREATEFIGURE(XDATA1,YDATA1,ZDATA1,XDATA2,YDATA2,ZDATA2,XDATA3,YDATA3,ZDATA3)
% XDATA1: line xdata
% YDATA1: line ydata
% ZDATA1: line zdata
% XDATA2: line xdata
% YDATA2: line ydata
% ZDATA2: line zdata
% XDATA3: line xdata
% YDATA3: line ydata
% ZDATA3: line zdata

% Auto-generated by MATLAB on 07-Oct-2008 11:39:34

% Create figure
figure1 = figure(‘NumberTitle’,’off’,’Name’,’Lorenz Attractor’,…
‘Color’,[0.35 0.35 0.35]);

% Create axes
axes1 = axes(‘Parent’,figure1,’ZTick’,zeros(1,0),’ZColor’,[1 1 1],…
‘YTick’,zeros(1,0),…
‘YColor’,[1 1 1],…
‘XTick’,zeros(1,0),…
‘XColor’,[1 1 1],…
‘Position’,[0.05 0.1 0.75 0.95],…
‘ColorOrder’,[1 1 0;1 0 1;0 1 1;1 0 0;0 1 0;0 0 1;0.75 0.75 0.75],…
‘Color’,[0 0 0]);
% Uncomment the following line to preserve the X-limits of the axes
% xlim([0 40]);
% Uncomment the following line to preserve the Y-limits of the axes
% ylim([-35 10]);
% Uncomment the following line to preserve the Z-limits of the axes
% zlim([-10 40]);
view([-37.5 30]);
hold(‘all’);

% Create xlabel
xlabel(‘X’,’Color’,[1 1 1]);

% Create ylabel
ylabel(‘Y’,’Color’,[1 1 1]);

% Create zlabel
zlabel(‘Z’,’Color’,[1 1 1]);

% Create line
line(XData1,YData1,ZData1,’Parent’,axes1,’MarkerSize’,25,’Marker’,’.’,…
‘Color’,[1 0 0]);

% Create line
line(XData2,YData2,ZData2,’Parent’,axes1,’Color’,[1 1 0]);

% Create line
line(XData3,YData3,ZData3,’Parent’,axes1,’Color’,[0 0 1]);

% uicontrol currently does not support code generation, enter ‘doc uicontrol’ for correct input syntax
% In order to generate code for uicontrol, you may use GUIDE. Enter ‘doc guide’ for more information

% uicontrol(…);

% uicontrol currently does not support code generation, enter ‘doc uicontrol’ for correct input syntax
% In order to generate code for uicontrol, you may use GUIDE. Enter ‘doc guide’ for more information

% uicontrol(…);

% uicontrol currently does not support code generation, enter ‘doc uicontrol’ for correct input syntax
% In order to generate code for uicontrol, you may use GUIDE. Enter ‘doc guide’ for more information

% uicontrol(…);

% uicontrol currently does not support code generation, enter ‘doc uicontrol’ for correct input syntax
% In order to generate code for uicontrol, you may use GUIDE. Enter ‘doc guide’ for more information

% uicontrol(…);

% uicontrol currently does not support code generation, enter ‘doc uicontrol’ for correct input syntax
% In order to generate code for uicontrol, you may use GUIDE. Enter ‘doc guide’ for more information

% uicontrol(…);

I have been trying to import our data and our images into a post. First from a page that I have been saving and adding to this entire project – they did not insert into the post – and now they are not in the page either. So, I then tried to add the media from my media library – again just a bunch of code appeared in the post. Hopefully I can figure this out soon. Just in case it takes longer than expected to find the data – I decided to write a short summary – I know that I could probably copy and paste from Matt’s Blog – I was there in class and you answered many questions along the way – but, I really want to try to figure it out on my own first.
I learned a lot from this project. It was a lot of fun and very interesting. I got answers a long the way to my questions … and discovered new questions that I am still looking for the answers to. We started out by following the guide and the example on the class web page.
We decided to start with the Lorenz system. We followed the example that was given. We used Excel to calculate our data using Euler’s method. It was very nice to use the computer because I got to see how different values for Delta affected the graph. Even the excel graphs were noticeably different. The computer just took seconds – so, it was easy to just play with the numbers and see what happens visually very quickly in a colorful way.
We had to work through some glitches with Excel and Matlab – when we tried to enter 8/3 into the cell – it gave us the date – we ended up just using the calculator feature and using 2.66.
Matlab is getting easier for me to navigate – finally 🙂 We did have some difficulties with things like loosing the command prompt and missing quotations. But, definitely not getting stuck as much.
Once we entered the code we got some really nice pictures of our Lorenz system. Again, I was curious about the different values for Delta and experimented at changing the viewpoint of the graphs too. How the system looked depended also on your viewpoint. A powerful connection to this – I learned later. I enjoyed being able to move the axis’.
I originally thought that we were basically done with the project early – but, I read through the example one more time. And, I noticed that there was one sentence about the system changing when P=99.96 so, I realized we were not finished – we needed to gather more data. The night before I had been doing some reading and researching on-line for a research topic and stumbled upon a research project about knots – that’s when everything seemed to connect and click for me. I had read about knots before – but, had “not” connected that – our system seemed to “Knot” ! I was really confused when we got the plot of (as Matt descibed ) a weird kind of “smooshed hershey kiss” shape. I thought I remembered from the lecture the first day of the project that when you start at a point in the system and follow it through – it doesn’t intersect its previous path – so, when I saw the graph starting and ending at the same point I got confused. And then I laughed when I learned it was the definition of a Knot. When I saw a 3D image of a knot it was very interesting – but it never appeared to intersect. Ha – since it didn’t have a starting and ending point – duh – ending and starting pt. the same!. A very basic simple concept – so, it was quite funny that it seemed like a light switch moment – the 2D connected with the 3D visually for me.
As we have learned that Euler’s method is not very accurate for non-linear systems – a very small error up front can grow to a very large margin of error at the end – kind of like the shape of a cornicopia – small (margin of error) beginning or opening – large(margin of error) ending or opening. I was also able to visually get a better understanding because of this lab. The 3D picture of the Lorenz system knotted – gave a width to the line ( but since 3D – more like a tube). So, I was able to see that we might think that we are really close with the approximation – but, not close enough – not only in 2D, but also in 3D.
So, in conclusion – I got to visually see why it is difficult to get an exact answer to a non-linear equation. And, exactly what all the math is doing and applications to the math that we are learning.

October 10, 2008

### Final Results of Project 1

September 19, 2008

Matt and I started working on the Differential Equation $\frac{dy}{dx}=(2y)/(x)$.
dsolve(‘DY = ((2*y)/(x))’, ‘y’)

ans =

y^2/x+C1

My Blog post on (9/9/08) reveals all the mistakes Matt and I made trying to get  direction field line graphs in Matlab.

(9/11/09)

Most of this week I was frustrated because I did not know or understand Matlab syntax.  It made it difficult to progress at the speed I wanted to.  Matt and I kept trying to figure out how to get the initial equation we had picked to graph.  We actually had picked this equation because it looked like the easiest to start with –  we wanted to complete the whole list of the equations before us.  Irony prevailed – as we spent a whole class time practically unable to figure out what we were doing wrong to graph in Matlab. … Thankfully, Matt suggested just trying another equation.  In less than twenty minutes we actually got some results !!!  Now we need to discuss the graghs of the solutions.  But, what I learned humored me – and, made me appreciate Matt and working in groups – most likely I would have just kept working at the same problem  – and brute forced it.  He was able to recognize that we needed to take a different approach – to be more effective – and that what we needed was to look at the assignment from a different angle.  I seemed to have forgotten this technique because I was lost in the problems we were having.  He helped remind me to step back and see the bigger picture.  To just shift the focus a little.  Over the past day and a half, I have been reading some primers on Matlab, and also reading one of the suggested texts.  Right now – it kind of feels disorderly – but, I am trying to press through the uneasiness to understand that learning isn’t necessarily linear – just like I learned from Matt this week – it is difficult to see a big picture if you are standing too close to it – so, as I try to step back – the goal will be to see everything as a whole and how the numerical calculations of Euler’s Method connect with the graphical behavior we are seeing with our “new” first equation!

This is our “new” first equation 🙂

>> SOL= dsolve(‘DT = -T*(Y+1)’, ‘T(0)=2′,’Y’)

SOL =

2*exp((-1/2*Y-1)*Y)

We were able to get some direction fields with this equation.  We tried to figure out what  the solution curves were.  The behavior of the slope lines did not seem to make any sense to us.  They looked like they were all pointing down at a large scale – but, they rotated at smaller scales around (.75) and lower.  I have never seen this before and cannot figure it out.

(graph 1)

(Matlab code)

>> [T,Y] = meshgrid(-1:0.1:1,-1:0.1:1);
>> S=-T*(Y+1);
>> L=sqrt(1+S.^2);
>> quiver(T,Y,1./L,S./L,0.5), axis tight (graph 1)

(graph 2) (Matlb code)
>> [T,Y] = meshgrid(-0.5:0.05:0.5,-0.5:0.05:0.5);
>> S=-T*(Y+1);
>> L=sqrt(1+S.^2);
>> quiver(T,Y,1./L,S./L,0.5), axis tight

2nd graph

(graph 3) (Matlab code)
>> [T,Y] = meshgrid(-0.1:0.01:0.1,-0.1:0.01:0.1);
>> S=-T*(Y+1);
>> L=sqrt(1+S.^2);
>> quiver(T,Y,1./L,S./L,0.5), axis tight

3rd graph

We tried to look at different intervals to see what behavior the solutions were taking …we started out trying to see the “big picture” with large increments – things got strange at very small increments … I am  not able to explain this at this time

This is a corrected direction field – this one matches our analytic results and is easier to see the behavior of the system.  Apparently we did not look in the right place on the plane.

>> [T,Y]=meshgrid(-5:.5:5, -5:.5:5);
>> S=-1*T.*(Y+1);
>> L=sqrt(1+S.^2);
>> quiver(T,Y, 1./L, S./L, 0.5),axis tight

I can see the solution curves on this graph – I am still not sure what is happening at (behavior) (-1)?  Is seems to behave like an asymptote.

Next, we used Matlab with different parameters to get a further qualitative look at our equation.

ezplot (SOL,[-.01, .01]), axis tight             (very small increments – look almost linear)

ezplot (SOL,[-1, 1]), axis tight   (at a larger increment we start to see a curve)

ezplot (SOL,[-10, 10]), axis tight (now we see the big picture at a larger scale)

(9/18/08)
Matt found another studen’t work (Joe Cross) on our same equation and was able to make some corrections to our work.  Today’s task was to work on Euler’s method.  I new we could use Excel to get very small increments for delta x – but, unsure how to get each cell to carry the calculations so that we did not have to input each increment manually.  Gary came over and showed us the method in Excel and even added some pretty colors to our graph.
This is Matt’s work …
Euler’s Method
$\frac{dy}{dx}=\frac{f(x+\Delta x)-f(x)}{\Delta x}$
Our equation here is $\frac{dy}{dx}= -x(y+1)$ and if we set the two equal to each other, we get:
$\frac{dy}{dx}=\frac{f(x+\Delta x)-f(x)}{\Delta x} = -x(y+1)$
$\frac{dy}{dx}=f(x+\Delta x)-f(x)= -x(y+1)*\Delta x$
$\frac{dy}{dx}=f(x+\Delta x)= f(x) -x(y+1)*\Delta x$
Excel’s Calculations of Euler’s Method – analytical findings to match the qualitative behaviors

delta x x y
0.1 1 5
1.1 4.4
1.2 3.806
1.3 3.22928
1.4 2.6794736
1.5 2.164347296
1.6 1.689695202
1.7 1.259343969
1.8 0.875255495
1.9 0.537709506
2 0.245544699
2.1 -0.00356424
2.2 -0.21281575
2.3 -0.385996285
2.4 -0.527217139
2.5 -0.640685026
2.6 -0.730513769
2.7 -0.800580189
2.8 -0.854423538
2.9 -0.895184948
3 -0.925581313
3.1 -0.947906919
3.2 -0.964055774
3.3 -0.975557926
3.4 -0.983623811
3.5 -0.989191715
3.6 -0.992974615
3.7 -0.995503753
3.8 -0.997167365
3.9 -0.998243766
4 -0.998928697
4.1 -0.999357218
4.2 -0.999620759
4.3 -0.99978004
4.4 -0.999874623
4.5 -0.999929789
4.6 -0.999961384
4.7 -0.999979147
4.8 -0.999988948
4.9 -0.999994253
5 -0.999997069
5.1 -0.999998535
5.2 -0.999999282
5.3 -0.999999655
5.4 -0.999999838
5.5 -0.999999925
5.6 -0.999999966
5.7 -0.999999985
5.8 -0.999999994
5.9 -0.999999997
6 -0.999999999
6.1 -1
6.2 -1
6.3 -1
6.4 -1
6.5 -1
6.6 -1
6.7 -1
6.8 -1
6.9 -1
7 -1
7.1 -1
7.2 -1
7.3 -1
7.4 -1
7.5 -1
7.6 -1
7.7 -1
7.8 -1
7.9 -1
8 -1
8.1 -1
8.2 -1
8.3 -1
8.4 -1
8.5 -1
8.6 -1
8.7 -1
8.8 -1
8.9 -1
9 -1
9.1 -1
9.2 -1
9.3 -1
9.4 -1
9.5 -1
9.6 -1
9.7 -1
9.8 -1
9.9 -1
10 -1
10.1 -1
10.2 -1
10.3 -1
10.4 -1
10.5 -1
10.6 -1
10.7 -1
10.8 -1
10.9 -1
11 -1
11.1 -1
11.2 -1
11.3 -1
11.4 -1
11.5 -1
11.6 -1
11.7 -1
11.8 -1
11.9 -1
12 -1
12.1 -1
12.2 -1
12.3 -1
12.4 -1
12.5 -1
12.6 -1
12.7 -1
12.8 -1
12.9 -1
13 -1
13.1 -1
13.2 -1
13.3 -1
13.4 -1
13.5 -1
13.6 -1
13.7 -1
13.8 -1
13.9 -1
14 -1
14.1 -1
14.2 -1
14.3 -1
14.4 -1
14.5 -1
14.6 -1
14.7 -1
14.8 -1
14.9 -1
15 -1
15.1 -1
15.2 -1
15.3 -1
15.4 -1
15.5 -1
15.6 -1
15.7 -1
15.8 -1
15.9 -1
16 -1
16.1 -1
16.2 -1
16.3 -1
16.4 -1
16.5 -1
16.6 -1
16.7 -1
16.8 -1
16.9 -1
17 -1
17.1 -1
17.2 -1
17.3 -1
17.4 -1
17.5 -1
17.6 -1
17.7 -1
17.8 -1
17.9 -1
18 -1
18.1 -1
18.2 -1
18.3 -1
18.4 -1
18.5 -1
18.6 -1
18.7 -1
18.8 -1
18.9 -1
19 -1
19.1 -1
19.2 -1
19.3 -1
19.4 -1
19.5 -1
19.6 -1
19.7 -1
19.8 -1

This is a graph of the Euler Method that we inputed in Excel – it matches our corrected direction field graph.

I found this project very difficult from the aspect of Matlab.  I was very thankful to have Mat as a partner.  I was a little frustrated because I did not feel like I was contributing to the level that I wanted.  But, it has caused me to push myself further – not just to be of better assistance to my partner – but, it helped me recognize what I did not know and need to know to keep up with my peers and better myself and my career.  I found that at first nothing seemed to connect and make sense – but the more I stayed with it – it all tied in at the end to make a “big picture” 🙂  Each portion added to the other – the qualitative and analytical was very helpful to me.

### Euler’s Method

September 18, 2008

delta x x y
0.1 1 5
1.1 4.4
1.2 3.806
1.3 3.22928
1.4 2.6794736
1.5 2.164347296
1.6 1.689695202
1.7 1.259343969
1.8 0.875255495
1.9 0.537709506
2 0.245544699
2.1 -0.00356424
2.2 -0.21281575
2.3 -0.385996285
2.4 -0.527217139
2.5 -0.640685026
2.6 -0.730513769
2.7 -0.800580189
2.8 -0.854423538
2.9 -0.895184948
3 -0.925581313
3.1 -0.947906919
3.2 -0.964055774
3.3 -0.975557926
3.4 -0.983623811
3.5 -0.989191715
3.6 -0.992974615
3.7 -0.995503753
3.8 -0.997167365
3.9 -0.998243766
4 -0.998928697
4.1 -0.999357218
4.2 -0.999620759
4.3 -0.99978004
4.4 -0.999874623
4.5 -0.999929789
4.6 -0.999961384
4.7 -0.999979147
4.8 -0.999988948
4.9 -0.999994253
5 -0.999997069
5.1 -0.999998535
5.2 -0.999999282
5.3 -0.999999655
5.4 -0.999999838
5.5 -0.999999925
5.6 -0.999999966
5.7 -0.999999985
5.8 -0.999999994
5.9 -0.999999997
6 -0.999999999
6.1 -1
6.2 -1
6.3 -1
6.4 -1
6.5 -1
6.6 -1
6.7 -1
6.8 -1
6.9 -1
7 -1
7.1 -1
7.2 -1
7.3 -1
7.4 -1
7.5 -1
7.6 -1
7.7 -1
7.8 -1
7.9 -1
8 -1
8.1 -1
8.2 -1
8.3 -1
8.4 -1
8.5 -1
8.6 -1
8.7 -1
8.8 -1
8.9 -1
9 -1
9.1 -1
9.2 -1
9.3 -1
9.4 -1
9.5 -1
9.6 -1
9.7 -1
9.8 -1
9.9 -1
10 -1
10.1 -1
10.2 -1
10.3 -1
10.4 -1
10.5 -1
10.6 -1
10.7 -1
10.8 -1
10.9 -1
11 -1
11.1 -1
11.2 -1
11.3 -1
11.4 -1
11.5 -1
11.6 -1
11.7 -1
11.8 -1
11.9 -1
12 -1
12.1 -1
12.2 -1
12.3 -1
12.4 -1
12.5 -1
12.6 -1
12.7 -1
12.8 -1
12.9 -1
13 -1
13.1 -1
13.2 -1
13.3 -1
13.4 -1
13.5 -1
13.6 -1
13.7 -1
13.8 -1
13.9 -1
14 -1
14.1 -1
14.2 -1
14.3 -1
14.4 -1
14.5 -1
14.6 -1
14.7 -1
14.8 -1
14.9 -1
15 -1
15.1 -1
15.2 -1
15.3 -1
15.4 -1
15.5 -1
15.6 -1
15.7 -1
15.8 -1
15.9 -1
16 -1
16.1 -1
16.2 -1
16.3 -1
16.4 -1
16.5 -1
16.6 -1
16.7 -1
16.8 -1
16.9 -1
17 -1
17.1 -1
17.2 -1
17.3 -1
17.4 -1
17.5 -1
17.6 -1
17.7 -1
17.8 -1
17.9 -1
18 -1
18.1 -1
18.2 -1
18.3 -1
18.4 -1
18.5 -1
18.6 -1
18.7 -1
18.8 -1
18.9 -1
19 -1
19.1 -1
19.2 -1
19.3 -1
19.4 -1
19.5 -1
19.6 -1
19.7 -1
19.8 -1

>> [T,Y]=meshgrid(-5:.5:5, -5:.5:5);
>> S=-1*T.*(Y+1);
>> L=sqrt(1+S.^2);
>> quiver(T,Y, 1./L, S./L, 0.5),axis tight