text:
User:Dpatel5391/enes100/Helicopter

Link to project Helicopter team page.

Give instructor top 3 project choice list
My top three project choices are Helicopter, Airplane Float, and Model boat. I am currently working on Helicopter project.

Write problem statement
In previous attempts, the Arduino code (C++) used by Masood Muhammad made it easier to add more functionality, including starting the motors with delay or at once. However, they were unsuccessful at writing a code that operates accelerometer and runs motors at once.

Link to their pictures: Previous Arduino Attempts

Assign Task1
The task I was assigned for Week1 was to learn how to use the Arduino software and download the required libraries to operate motors and accelerometer.

Compare actual work done to Task1
This task was very useful, since now I know how to write and edit a sketch. The work I completed in Task1 was really helpful to me in understanding how the Arduino software works by using C++ codes and libraries. Using the Arduino with my computer helped me understand how writing different codes or making mistakes changes its functionality.

Week1 Narrative
I have downloaded the Arduino Software, you can find the windows version at "WinOS-Downloads". The version I have downloaded is 0022, which is the latest stable release as of today. Before I started to use the software, I needed to learn how to write the sketches, such as C++ codes and libraries. The following are five of the most important guides to learn the software basics.

 * Arduino Development Environment: This is a manual on using Arduino, which explains how to use a text editor for writing code, a message area, a text console, series of menus and toolbars, and also explains how to upload your sketch step by step.

 * Jeremy Blum's video guide: This is a video guide, in which Jeremy explains what an arduino is and goes through the downloading process, materials needed, running the program, writing the code, and uploading it onto the arduino. He has uploaded a total of 13 tutorial videos, in which he guides us through various functionalities of Arduino by writing different, specific codes for each of them. Those videos might be helpful to me in later activities.

 * Arduino Troubleshooting, Archive for the ‘Arduino’ Category, and Correcting Arduino Compiler Errors: These tutorials go over many different errors that we might get when trying to program arduino and show how to resolve them.

I am currently working with three devices, Arduino-UNO, Arduino breadboard, and Accelerometer ADXL335. I couldn't get to work with the accelerometer and the breadboard, since I kept on getting an error when trying to upload the sketch (I used the code from Jeremy Blum's video guide /10:15/ for a test). Everytime, I tried to upload this sketch I kept on getting an error message saying:

Main Error: function 'void setup' is initialized like a variable

sketch_sep25a:10: error: function 'void setup' is initialized like a variable

sketch_sep25a:11: error: expected qualified-id before ')' token

sketch_sep25a:12: error: expected constructor, destructor, or type conversion before '(' token

To correct these errors, I used the link Correcting Arduino Compiler Errors /Line 12 Error/, which explains that I made a mistake when entering the code where I needed to use curly brace "{, }" instead of parenthesis. I corrected my mistake; however, I had another mistake where I forgot to capitalize the letter P in "ledPin." After I corrected all of my mistakes, I successfully uploaded the sketch onto the arduino.

Here's the picture and my sketch of how it should look like after you're done uploading it.

There's also another way to do the same functionality, but in this case the software itself writes the code, which you can upload and don't have to worry about getting any errors.

1) Run the Arduino Software 0022, 2) under File menu, go to Examples > Basics > Blink > Upload.

Besides doing these activities, I also went to College Park Airport located onto 6709 Corporal Frank Scott Dr. for helicopter tail configuration. Mr. Paul had two air crafts disassembled for inspection. For one of them he was working on the main rotors and for the second one he was working on the tail and the engine. He took out the tail and made it easier for me to document the configuration of it and here're some pictures I took. It was really a great experience to see the helicopter and learning about it. From Mr. Paul, I learned so much about why the helicopters are designed in specific ways.

Week1 Peer Review
My comment was regarding to how Matt should share his work with group members.

Assign Task2
I am planning to connect the Arduino with my computer again via USB cable, but this time I am going to upload different types of sketches to run motors using Motor Control Shield for Arduino.

Compare actual work done to Task2
This task was really time consuming, yet interesting, fun, and one of my favorites. The work I completed in Task2 was really helpful to me in understanding how to write different sketches to run four DC-motors using the ports M1, M2, M3, and M4; and also in determining if I was able to run all motors either in the same or different directions at once.

Week2 Narrative
I have downloaded the |944l3|526933304|AFMotor.rar|6|R~D7E9166CDDFC24E3D4BA8A3905932087|0|0 AFMotor library from http://www.ladyada.net/make/mshield/download.html, where you can also download Arduino Stepper/Servo software library with microstepping support and AccelStepper library with AFMotor support. Before I start using the AFMotor library, I needed to know what the library is used for, how it works, and how to install it. The following are three of the most important guides to learn about the libraries.

Arduino Hacks: Libraries. This is a manual on using different types of libraries, which basically explains how to install, import, and use the libraries.

Jeremy Blum's video guide. This is a video guide, in which Jeremy explains and talks about NPN Transistor and PWM, which allows to run a motor faster through arduino. He also talks about using the different motor libraries, and writing appropriate sketches when using different motors, such as DC-motors, survo and stepping motors.

Motor Shield. This tutorial includes powering the DC motors, voltage and current requirements for the first section. The second section is about how to set up the arduino and the motor control shield for powering motors, and the remaining sections cover Survos, DC-motors, and Steppers. The last two sections include step by step directions on how to run the motors using the sketches they have provided. I have tested those sketches and they seem to work very well.

Note: if you were to download the AFMotor library I have provided. Follow these steps:

1) Download and extract the files onto your Desktop.

2) Copy the AFMotor folder and paste it into your existing folder: Arduino-0022/libraries/

Now, you can easily import the library by going to Sketch > Import library > AFMotor. You can also import the motor sketches by going to File > Examples > AFMotor

I am currently working with six devices, including Arduino-UNO, the breadboard, motor control shield, DC-motors, stepping motors, and AC power adapter for arduino. Here's the overall picture of how I connected the Arduino UNO, the motor control shield and the DC-motor.

When trying to upload the first Motor-test sketch, I kept on getting an error message

Problem uploading to board.

Binary sketch size: 3318 bytes (of a 32256 byte maximum)

avrdude: stk500_getsync: not in sync: res=0x30

avrdude: stk500_getsync: protocol error, expect=0x14, resp=0x51

To correct this error, I used the link Arduino Troubleshooting, which was already provided under the error window. There are many questions listed on the page, where I was more interested in "Why do I get errors about an invalid device signature when trying to upload a sketch?" The only answer to this question was that either you have the wrong board selected from the Tools > Board menu or you're not using the right version of avrdude. The problem with me was that I selected the wrong board-COM3 instead of COM4. Now I was able to upload the sketch and run the motor through M4-port of the motor control shield.

There're two ways you can change the directions in which the motors spin. One way is to write a sketch that spins the DC-motors either only in "FORWARD" or "BACKWARD" direction. Another way is to change the order of the wires connected to the ports of the motor control shield (I mentioned about it in My Video). This basically means the motors would start spinning by the time we plug the wires into the ports (M3 & M4) and stops when taking out of the ports. If I want to run the motors in "FORWARD" direction, I would delete the parts which I don't need, including the variables "BACKWARD (reverse)," and "RELEASE (stop)." I used the yellow and red LEDs to test whether the M3 and M4 are both working at the same time or not. Finally, I came up with the Sketch I was looking for and here's the picture showing the two LEDs turned on at once.

Now, it was the time to write a sketch that spins four DC-motors at once; however writing the sketch was pretty easy and straight forward, since all I had to do was to add six more variables to the previous sketch. Here're the changes I made to the previous sketch to get the    Final Sketch:

First I added:

AF_DCMotor motor3(3);

AF_DCMotor motor4(4);

Then under 'void setup' I added:

motor3.setSpeed(200);

motor4.setSpeed(200);

and under the 'void loop' I added:

motor3.run(FORWARD);

motor4.run(FORWARD);

The problem which I was most concerned about was that when I tested the connection of each ports using LEDs, they were all working good (here's My Video); however, when I connected all four motors to it, I saw that only three of the motors were working and one of them didn't even spin once. Here's the Video I took explaining the problem and realizing that the power shortage was the main issue. The problem was fixed when I connected the arduino to AC power adapter; however the power was going back and forth so the motors were also functioning in the same way.

Week2 Peer Review
My comment was again regarding to how Matt should share his work with group members.

Assign Task3
I am planning to continue with the Arduino UNO, and the motor shield but this time I am also going to use the Accelerometer ADXL335 and upload different types of sketches to run steppers and DC-motors all at once using external power supply.

Compare actual work done to Task3
This task was also time consuming, yet exciting. The work I completed in Task3 was really helpful to me in understanding how the accelerometer ADXL335 works using Prototype with whitebreadboard attached to it and all of these attached to the Arduino UNO. It also helped me understand how you can actually run DC-motors and stepping motors at once in which the two DC-motors uses M1 and M2 ports, where as the only one stepping motor uses two ports, M3 and M4. I didn't get to learn much about Accelerometer ADXL 335 and I admit that it's really complicated than I expected.

Week3 Narrative
For this task, I didn't have to download anything since I already have everything I needed from the previous tasks. Like wise, I also had the Stepper librarary which was included in the AFmotor library. Since past two weeks, I've been working with the arduino, motor control shield, whitebread board, and the DC motors. Now it's the time to learn about the Accelerometer ADXL335 and the stepping motor. Here are the links to guides and basic explanation of each guide.

Analog Devices: This is one of the most important guides, which explains everything you need to know about the Accelerometer ADXL335. The link goes through general description of the accelerometer, functional block diagram, specifications, Pin configuration and function description, typical performance characteristics, Mechanical Sensor, Applications information (such as Power Suply Decoupling, Self-Test, Setting the Bandwidth using CX, CY, and CZ), Axes of Acceleration Sensitivity, and Outline its three dimensions (X,Y, and Z). Motor Shield (stepper): This is a guide to learn about stepper motor. It explains how Bi-polar steppers are different from Uni-polar steppers and their differences in writing sketches. There's also a picture of a Bi-polar stepper (4 wires). It also provides a Test-Sketch for it.

Circuit for Bipolar Stepper Motor: This is a diagram of how Bi-polar steppers are different from two pins to four pins.

Circuits for Unipolar Stepper Motors: This is a diagram of how Unipolar steppers are different from two pins to four pins.

Driving a Unipolar Stepper Motor: This tutorial shows how to write a sketch for Unipolar Stepper and run it.

The main difference between DC motors and steppers is that the motor control shield can drive up to 4 DC motors bi-directionally, where as it supports only up to 2 stepper motors. Running a stepper is a little more intricate than running a DC motor. I already have a sketch for running 4 DC motors at once, and now I'm interested in running a stepper motor with 3 DC motors. Here's the list of devices I used for this task:

Arduino UNO

Accelerometer ADXL335

Motor Control Shield

Prototype with Whitebread board

Three DC motors

One Stepping motor

Here's the overall picture of how it should look like after you have attached everything together. (Note: you won't be able to attach the Motor Control Shield and the Prototype to the arduino at the same time.) Picture of Motor Control Shield attached to the arduino, and Picture of Prototype attached to the arduino.

Now, I am more interested in writing a sketch that will run a stepper motor. For the first trial, I used the sketch provided by the AFmotor library. Here's how to access the sketch: Run the arduino software > File > Examples > AFmotor > Stepper motor, then upload this sketch onto the arduino. Once you have uploaded the sketch, Here's how to connect the wires of a Bi-polar stepper motor (4 wires): You have total of 4 wires to a stepper motor. (Note: No matter which side you choose, the connection to the ports is always the same.) The first wire goes into the first port of M3, the second wire goes into the first port of M4, the third wire goes into the second port of M3, and the fourth wire goes into the second port of M4. Here's the Picture of it. As soon you attach the last wire, the motor should start spinning. Here's the overall Picture of how it should look like when you're finished connecting the wires.

Now, I see that the motor spins in both directions; however I want the motor to spin only in one direction as DC-motors in Task2. I also want to add the DC-motors to it, so I can run all at once. So here's how I edited the Sketch. Here's the Picture of it. Hit Next for the other half of the sketch. I have also uploaded my video showing how to run 2 or more DC motors when running with 1 stepper motor; however I couldn't get to run more than 3 DC motors with the stepper motor due to power shortage.

After completing my task for the stepper motor, I started working on the Accelerometer ADXL335.

What is Accelerometer ADXL335? "The ADXL335 is a small, thin, low power, complete 3-axis accelerometer with signal conditioned voltage outputs. It measures acceleration with a minimum full-scale range of ±3 g. It can measure the static acceleration of gravity in tiltsensing applications, as well as dynamic acceleration resulting from motion, shock, or vibration." Applications for which I will need the Accelerometer ADXL335 are Cost sensitive, low power, motion- and tilt-sensing. The motor also creates different sounds which you can refer to for changing the directions of it. Here's a basic Diagram of Accelerometer ADXL335 and Picture of how to connect the Accelerometer ADXL335 to the Arduino UNO. Learning and writing the Accelerometer sketch by self is far more complicated than using a tested-working sketch provided by arduino professionals. Here's the Picture of the sketch I used to test the Accelerometer ADXL335. Hit Next three times to see the complete view of the sketch. Click Here to download the sketch. Here's a Picture of how the accelerometer works using hand coordination and tilting in different directions.

When I was doing an LED-test for the connection of the ports from Prototype while using the Sketch, here's what I found out:

Regular LED Pin#13 - light is off (Note: It only flashes when I hit the "Reset" button).

v6 Pin (next to GND)- light is normal (Red for a red LED)

All connections from v5 of Prototype Board - light is orange (Note: No matter which LEDs I use, the only outcome is an orange light. I tested it by using yellow, green and red LEDs).

Every ports next to Accelerometer ADXL335 - Light is normal; however I accidently put both wires of the yellow LED into the GND ports and the light just went off and didn't work at all. I also smelled something burning from the Whitebread Board, so I suddenly stopped working and thought of testing the ports using different methods. Picture when I was testing the connection going through different ports when using the prototype and accelerometer. If you hit Next arrow, there're 2 more pictures to it.

Here's a Picture of a motor with the helicopter rotors that I made by hand from scratch. I was looking at this Flour Bag that was attached to a red plastic handle. Then, I took out the handle and cut from both sides equally. Then, I attached those two pieces together by burning them with the remaining plastic strip while attached to the motor. I let it cool down for 15 minutes and then attached the motor to arduino to test and it worked very well.

Week3 Peer Review
My comment was still regarding to how Matt should share his work with group members.

Assign Task4
I've found many tutorials to control the stepper motor speed using variable resistors through motor control shield but haven't found any tutorial that clearly specifies how to control the speed and change the direction of a motor using the Accelerometer ADXL335. I am planning forward to use the accelerometer ADXL335 with the Arduino UNO and have a remote controller speeding up, slowing down, and changing the direction of a motor.

Compare actual work done to Task4
The work I completed in Task4 was really helpful to me in understanding how the accelerometer ADXL335 works with prototype shield. There are only limited sketches available to work with accelerometers, and also few tutorials which explain how to write a sketch to program an accelerometer. The task involved many failures due to use of different types of variables and keeping track of each variable and its function.

Week4 Narrative
For this task, I also didn't have to download anything, since the accelerometer library is already included in the software files. I started Task4 by researching information about Accelerometer ADXL335 and prototype shield. I found the following tutorials which I thought were very important.

 * Playing with an Arduino and sensors: is one of the best tutorials available in details and clearly explains all the topics related to any accelerometer. The tutorial includes total of 11 contents which are:
 * 1)  The Hardware;
 * 2)  First software: reading the IMU data;
 * 3)  Getting rid of zero bias, calculating angles for pitch and roll, and a mysterious foo factor;
 * 4)  The mathematically painful bit: the Kalman filter
 * 5)  Intermezzo: an idea for an artificial horizon
 * 6)  What about yaw?
 * 7)  Another intermezzo: alternative hardware
 * 8)  Vertical Acceleration Rejection
 * 9)  Cuidado! Lama!
 * 10) The controller
 * 11) Some filtering required

 * Sensing Orientation With The ADXL335 + Arduino: is a basic tutorial that goes through two main topics: 1) "Basic Understanding Of Accelerometers" and 2) "Code For The ADXL335 Or Other Analog Accelerometer." This link also provides a code which you may want to test to see if it works. I tried to test it, but I wasn't sure about how to setup everything in order for the code to work. Here's the code:

 * Here is another sketch I found that I thought was really helpful.

I was really interested in using the sketch; however, I couldn't get the setup correctly. Every time I tried tilting the accelerometer, but nothing worked. I even tried performing an LED test for both of the sketches, but the lights either stayed on or off and didn't function properly.

Week4 Peer Review
My comment was again regarding to how Matt should at least document some of his work.

Start Next Project Week0 activities
I am planning to keep on working over the same project, since I am getting so close to programming a Quadcopter. For the next step, I am going to test the accelerometer using 5 LEDs, and see how the sensor indicates any change in motion, and also planning to write a sketch so that I will be able to run or turn off a motor by using a switch.

title: User:Dpatel5391/enes100/Helicopter


text:
User:Dmoreno1899/Enes100/Music Project

Link to project MusicTeam team page.

Give instructor top 3 project choice list
Typically the instructor picks teams. Students send the instructor their top 3 choices. The project choices are typically those found at Category:General_Engineering_Projects. The instructor tries to put students in the project of their choice. Put your top three choices here after they are used to assign you to this project. Create a link to the project root and to where your team will be working.

Write problem statement
Look through the existing problem statements. Find something not yet done on the project. Create a problem statement for it. Think of a new subsystem that needs to be built for the project. The goal is something simple, possible, for zero dollars.

Assign Task1
Tasks are typically done over the weekend. They are documented during the week, presented during the week and the next weekends work is negotiated during the week. Week 0 is the week that task 1 is negotiated. Record your individual task here. Plan on working on it during the weekend between week0 and week1.

Compare actual work done to Task1
Rationalize differences

Week1 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
Record what you are planning on doing for the team during the weekend between week1 and week2 of the project here.

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Dmoreno1899/Enes100/Music Project


text:
User:MScobell1809/enes100/Helicopter

Link to project Helicopter team page.

Give instructor top 3 project choice list
Typically the instructor picks teams. Students send the instructor their top 3 choices. The project choices are typically those found at Category:General_Engineering_Projects. The instructor tries to put students in the project of their choice. Put your top three choices here after they are used to assign you to this project. Create a link to the project root and to where your team will be working.

Write problem statement
Look through the existing problem statements. Find something not yet done on the project. Create a problem statement for it. Think of a new subsystem that needs to be built for the project. The goal is something simple, possible, for zero dollars.

Assign Task1
Tasks are typically done over the weekend. They are documented during the week, presented during the week and the next weekends work is negotiated during the week. Week 0 is the week that task 1 is negotiated. Record your individual task here. Plan on working on it during the weekend between week0 and week1.

Compare actual work done to Task1
Rationalize differences

Week1 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
Record what you are planning on doing for the team during the weekend between week1 and week2 of the project here.

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:MScobell1809/enes100/Helicopter


text:
User:Koalex/poppit

Link to project 1+1=11 team page.

Give instructor top 3 project choice list
Typically the instructor picks teams. Students send the instructor their top 3 choices. The project choices are typically those found at Category:General_Engineering_Projects. The instructor tries to put students in the project of their choice. Put your top three choices here after they are used to assign you to this project. Create a link to the project root and to where your team will be working.

Write problem statement
Look through the existing problem statements. Find something not yet done on the project. Create a problem statement for it. Think of a new subsystem that needs to be built for the project. The goal is something simple, possible, for zero dollars.

Assign Task1
Tasks are typically done over the weekend. They are documented during the week, presented during the week and the next weekends work is negotiated during the week. Week 0 is the week that task 1 is negotiated. Record your individual task here. Plan on working on it during the weekend between week0 and week1.

Compare actual work done to Task1
Rationalize differences

Week1 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
Record what you are planning on doing for the team during the weekend between week1 and week2 of the project here.

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:Koalex/poppit


text:
Engineering Projects/Poppit/Howard Community College/Fall2011/502 ABM

Problem Statement
Put the last negotiated problem statement with the instructor or client of the project here.

Team Members
Put each team members name here with links to their personal page here. Real names do not have to be used.

Summary
Put an overall, one paragraph summary here with links to the team weekly reports.

Poster
Put a graphic in wikimedia, include the graphic here or link to it here. The graphic should be suitable for creating a traditional project poster.

Story
Tell a story of the project. Describe how split up, what the obstacles were, what testing was done, what informal decisions were made, what assumptions were made, what the results were. This is a longer version of the summary with links to all the details collected associated with the project. These links could be to software, links to videos, links to project pages with pictures, etc. Think of the story as a summary of the team weekly reports on one long page rather than a short paragraph like the summary.

Decision List
List all formal decisions made with links to their documentation such as a decision tree or decision matrix.

Material List
Figuring out what to purchase, what to build, what everything costs is a huge part of engineering. Typically there is a list of materials in stock, materials that are ordered, materials that should be ordered next time there is money, materials that have not been fully justified. These issues are part of healthy management of the engineering lab but are associated with a particular college. The detail needs to be published in this "Done" form. However in the project root, just a list of materials is necessary.

Software List
Installing and learning different software packages is part of most engineering project just like the materials list.

Time
Time estimates and actual time consumed measurements helps justify salaries (grades). The only way to gain respect for estimated project time and costs is to practice.

Tutorials
Most projects consist of making instructions to jump start the next team, shrink the learning of tools and software to the minimum, advice on where to purchase materials, how to assemble, etc. Tutorials modified or created are described here with links.

Next Steps
If this is not filled out, the next team may have to repeat each of your steps just to figure out what the next step is. It is best to write this immediately at the end of the project while creating the summary, poster, presentation, and story.

Subjective Critera
While reading these sections, practicing engineers are building up the following impressions about the document. So the following categories are more subjective:

Repeatablitly
Is there enough detail to repeat the work?

Integrity
Proving something doesn't exist or doesn't work usually requires exhaustion. Simply saying "it can not be done" is like challenging a pro basketball player (engineer) to one-on-one basketball (find a solution). It does not inspire respect. All the different ways respect is created can be found throughout the wikibook General Engineering Introduction.


text:
User:Egm6321.f11.team1/report2

= R*2.1 Verify the solution of the specific Legendre differential equation=

Given
The Legendre differential equation is
 * $$\displaystyle L_2(y):=(1-x^2)y''-2xy'+n(n+1)y=0$$ .

For the specific case $$\displaystyle n=1$$ , the Legendre differential equation becomes
 * {| style="width:100%" border="0" align="left"

$$ The two homogeneous solutions are given:
 * $$\displaystyle (1-x^2)y''-2xy'+2y=0$$
 * $$\displaystyle (Eq.2.1.1)
 * $$\displaystyle (Eq.2.1.1)
 * }
 * }
 * $$\displaystyle y_H^1(x)=x$$ ;
 * $$\displaystyle y_H^2(x)=\frac{x}{2}log\left(\frac{1+x}{1-x}\right)-1$$ .

== Find == Verify that
 * $$L_2(y_H^1)=L_2(y_H^2)=0$$ .

Solution
For $$y_H^1$$ ,  its first derivative is
 * $$\displaystyle \frac{dy_H^!}{dx}=1$$ ;

and its second derivative is
 * $$\displaystyle \frac{d^2y_H^1}{dx^2}=0$$ .

Take them into $$\displaystyle (Eq.2.1.1)$$ ,

 * $$\displaystyle L_2(y_H^1)=(1-x^2)\cdot 0-2x\cdot 1+2\cdot x=0$$ .

For $$y_H^2$$ ,

 its first derivative is

 * $$\displaystyle \frac{dy_H^2}{dx}=\frac{1}{2}log\left(\frac{1+x}{1-x}\right)+\frac{x}{2}\cdot\frac{1-x}{1+x}\cdot\left(\frac{1+x}{1-x}\right)'$$

 * $$\displaystyle =\frac{1}{2}log\left(\frac{1+x}{1-x}\right)+\frac{x}{1-x^2}$$ ;

 and its second derivative is

 * $$\displaystyle \frac{d^2y_H^2}{dx^2}=\frac{1}{2}\cdot\frac{1-x}{1+x}\cdot\frac{2}{(1-x)^2}+\frac{1-x^2+2x^2}{(1-x^2)^2}$$

 * $$\displaystyle =\frac{1}{1-x^2}+\frac{1+x^2}{(1-x^2)^2}$$ .

Take them into $$\displaystyle (Eq.2.1.1)$$ ,

 * $$\displaystyle L_2(y_H^2)=(1-x^2)\cdot \left[\frac{1}{1-x^2}+\frac{1+x^2}{(1-x^2)^2}\right]-2x\cdot \left[\frac{1}{2}log\left(\frac{1+x}{1-x}\right)+\frac{x}{1-x^2}\right]+2\cdot \left[\frac{x}{2}log\left(\frac{1+x}{1-x}\right)-1\right]$$

 * $$\displaystyle =1+\frac{1+x^2}{1-x^2}-x\cdot log\left(\frac{1+x}{1-x}\right)-\frac{2x^2}{1-x^2}+x\cdot log\left(\frac{1+x}{1-x}\right)-2$$

 * $$\displaystyle =0$$ .

 Solution by EGM6321.f11.team1.pan

= R*2.2 Verify the solution of an equation= == Find == Verify that

 * $$\displaystyle p(x)=k_1e^{-x}+x-1$$

is indeed the solution for

 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle p'+p=x$$.
 *
$$\displaystyle (Eq.2.2.1)
 *
$$\displaystyle (Eq.2.2.1)
 * }
 * }

Solution
Derivate of $$\displaystyle p(x)=k_1e^{-x}+x-1$$ is

 * $$\displaystyle p'(x)=-k_1e^{-x}+1$$ .

Substituting its into $$\displaystyle (Eq.2.2.1)$$ ,

 * $$\displaystyle p'(x)+p(x)=-k_1e^{-x}+1+k_1e^{-x}+x-1=x$$ .

 Solution by EGM6321.f11.team1.pan

= R*2.3 Show that the equation is linear in $${y}'$$ but an N1-ODE in general = Show that (2)p.7-6 is linear in y', and that (2) is in general an N1-ODE. But (2） is not the most general N1-ODE as represented by (1)p.7-6. Give an example of a more general N1-ODE.

Given
General N1-ODEs:
 * {| style="width:100%" border="0"

$$  \displaystyle G({y}',y,x) =0 $$     (3-1) (1)p.7-6 Particular class of N1-ODEs: Linear in y'
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle M(x,y)+N(x,y){y}' =0 $$     (3-2) (2)p.7-6
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

== Find == 1)Show that (2)p.7-6 is linear in y' 2)that (2) is in general an N1-ODE. 3)Give an example of a more general N1-ODE.

1) Show that (2)p.7-6 is linear in y'
For a operator to be linear, it has to satisfy the following:
 * {| style="width:100%" border="0"

$$  \displaystyle F(\alpha u+\beta v)=\alpha F(u)+\beta F(v) ,\forall \alpha ,\beta \in \mathbb{R} $$     (3-3) Then we begin to check, Let Y = y',thus
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle G(x,y,Y)=\underbrace{M(x,y)}_{(1)}+\underbrace{N(x,y)Y}_{(2)} $$     (3-4) for (1)  We can regard this as a constant M; for (2)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle F(x,y,Y)=N(x,y)Y $$     (3-5)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle F(x,y,\alpha u+\beta v)=N(x,y)(\alpha u+\beta v)=\alpha N(x,y)u+\beta N(x,y)v $$     (3-6)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \alpha F(x,y,u)+\beta F(x,y,v)=\alpha N(x,y)u+\beta N(x,y)v $$     (3-7) So, Formula 3-6 is equal to 3-7, i.e.:
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle F(x,y,\alpha u+\beta v)=\alpha F(x,y,u)+\beta F(x,y,v) $$     (3-8) Then we can say that 3-2 i.e. (2)p.7-6 is linear in Y = y'
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

2) Show that (2) is in general an N1-ODE
For a operator to be linear, it has to satisfy the following:
 * {| style="width:100%" border="0"

$$  \displaystyle f(\alpha y_1+\beta y_2)=\alpha f(y_1)+\beta f(y_2) ,\forall \alpha ,\beta \in \mathbb{R} $$ Then we begin to check, So,
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle f(x,\alpha y_1+\beta y_2)\neq \alpha f(x,y_1)+\beta f(x,y_2) $$     (3-12) CONCLUSION: (2) is in general an N1-ODE
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 *

 * }

3) Give an example of a more general N1-ODE
(2） is not the most general N1-ODE as represented by (1)p.7-6.
 * {| style="width:100%" border="0"

$$  \displaystyle G(y',y,x)=0 $$     (3-13) A more general N1-ODE example can be:
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle M(x,y)+N(x,y)(y')^n=0 $$     (3-14) n is the power of y' This problem is solved by ourselves Contributed by Zexi
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

= R*2.4 Verify that $$(4x^7+\sin y)+(x^2y^3)y'=0$$ is a N1-ODE =

Given

 * {| style="width:100%" border="0"

$$  \displaystyle (4x^7+\sin y)+(x^2y^3)y'=0 $$     (4-1) == Find == Formula 4-1 is a N1-ODE
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Solution
1) The highest order of derivative is 1 and there is only one dependent variable in equation so it's a X1-ODE. (X can be L or N) 2) Verify Non-linearity For a operator to be linear, it has to satisfy the following:
 * {| style="width:100%" border="0"

$$  \displaystyle F(\alpha u+\beta v)=\alpha F(u)+\beta F(v) ,\forall \alpha ,\beta \in \mathbb{R} $$     (4-2) Then we begin to check,
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle F(x,y)=(4x^7+\sin y)+(x^2y^3)y' $$     (4-3)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle F(x,{\alpha}{y_1}+{\beta}{y_2})=4x^7 + sin({\alpha}{y_1}+{\beta}{y_2}) +{x^2}({\alpha}{y_1}+{\beta}{y_2})^3({\alpha}{y_1}+{\beta}{y_2})' $$     (4-4)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle {\alpha}F(x,{y_1})+{\beta}F(x,{y_2})={\alpha}(4x^7+sin{y_1}+x^2{y_1}^3{y_1}')+{\beta}(4x^7+sin{y_2}+x^2{y_2}^3{y_2}') $$     (4-5)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle F(x,{\alpha}{y_1}+{\beta}{y_2})\neq {\alpha}F(x,{y_1})+{\beta}F(x,{y_2}) $$     (4-6) So, it's nonlinear. CONCLUSION: The equation is a N1-ODE This problem is solved by ourselves Contributed by Zexi
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 *

 * }

= R*2.5 Conversion to Particular N1-ODE =

== Find ==

Explain the following equations:

 * {| style="width:100%" border="0" align="left"

$$
 * $$\bar M(x,y)+\bar N(x,y)(y')^3=0$$
 *
$$\displaystyle (Equation\;2.5.1)
 *
$$\displaystyle (Equation\;2.5.1)
 * }
 * }

 * $$M(x,y):=\left [\bar M(x,y)\right]^{1/3}$$

 * $$N(x,y):=\left [\bar N(x,y)\right]^{1/3}$$

Basic Equations
General N1-ODE form:

 * $$G\left(y',y,x\right)=0$$

Particular N1-ODE form:

 * {| style="width:100%" border="0"

$$M\left(x,y\right)+N(x,y)y'=0$$
 *
$$\,(Equation\;2.5.2) $$
 * style= |
 * }

Solution
The relation among the three equations we are to explain is illustrated by the ability to convert the first from a GENERAL form N1-ODE to a PARTICULAR form N1-ODE. The following steps show how redefining M and N as given will convert one type of equation into another.

Use definition of M and N and take cube power of each side.

$$ \left[M(x,y)\right]^{1/3}=\bar M(x,y) $$

$$ \left[N(x,y)\right]^{1/3}=\bar N(x,y) $$

Enter this result into Equation 2.5.1.

$$\left[M(x,y)\right]^3+\left[N(x,y)\right]^3(y')^3=0$$

Subtract one term to the right side.

$$\left[M(x,y)\right]^3=-\left[N(x,y)\right]^3(y')^3$$

$$\left[M(x,y)\right]^3=\left[-\left[N(x,y)\right](y')\right]^3$$

Take the cube root and add to the left side.

$$M(x,y)=-\left[N(x,y)\right](y')$$

$$M\left(x,y\right)+N(x,y)(y')=0$$

This now matches Equation 2.5.2. We can see that we were able to convert an apparent general form N1-ODE to a particular form N1-ODE.

 Solution by Egm6321.f11.team1.colsonfe 02:31, 17 September 2011 (UTC)

= R*2.6 Linear Independence of Homogeneous Solutions =

Given
Two homogeneous solutions:

 * $$y^1_H(x)=x$$

 * $$y^2_H(x)=\frac{x}{2}\log\left(\frac{1+x}{1-x}\right)-1$$

== Find ==

Show that $$y^1_H(x)$$ and $$y^2_H(x)$$ are linearly independent.

Plot $$y^1_H(x)$$ and $$y^2_H(x)$$.

Basic Equations
Linear Independence:

 * $$\forall\alpha\in\mathbb R, y^1_H(\cdot)\ne\alpha y^2_H(\cdot)$$

i.e, for any given $$\alpha$$, show that

 * $$\exists\hat x$$ such that $$y^1_H(\hat x) \ne\alpha y^2_H(\hat x)$$

Solution
By definition of linear independance, we merely need to show there is a value $$\hat x$$ where $$y^1_H(\hat x) \ne\alpha y^2_H(\hat x)$$ for some $$\displaystyle\alpha$$.

Let us then choose $$\hat x=0$$.Then

$$y^1_H(\hat x)=\hat x=0$$

$$y^2_H(\hat x)=\frac{\hat x}{2}\log\left(\frac{1+\hat x}{1-\hat x}\right)-1=-1$$

Now lets consider any given $$\displaystyle\alpha$$ for our chosen $$\hat x=0$$,

$$y^1_H(\hat x) \overset{?}{=}\alpha y^2_H(\hat x)$$

$$0 \overset{?}{=}-\alpha$$

It is clear that the above equality IS NOT true for all $$\displaystyle\alpha$$.

Therefore, for $$\hat x=0$$ and $$\alpha=1$$, $$y^1_H(\hat x) \ne\alpha y^2_H(\hat x)$$ for some $$\displaystyle\alpha$$, which means that $$y^1_H(x)$$ and $$y^2_H(x)$$ are linearly independent.

 * HW2_6plots.gif

 Solution by Egm6321.f11.team1.colsonfe 02:31, 17 September 2011 (UTC)

= R*2.7 Find G(y',y,x)=0, and verify it is an N1-ODE=

Given
Function $$\displaystyle \phi(x,y)=x^2y^{\frac{3}{2}}+log(x^3t^2)=k $$ .

== Find == Find $$\displaystyle G(y',y,x)=\frac{d}{dx}\phi(x,y)=0$$ .
 And show that this is an N1-ODE.

Solution
According to the Chain Rule,

 * $$\displaystyle \frac{d\phi(x,y)}{dx}=\frac{\partial\phi(x,y)}{\partial x}+\frac{\partial\phi(x,y)}{\partial y}\frac{dy}{dx}$$

 * $$\displaystyle =2xy^{\frac{3}{2}}+3x^2y^2\cdot\frac{1}{x^3y^2}+y'\left(\frac{3}{2}x^2y^{\frac{1}{2}}+2x^3y\frac{1}{x^3y^2}\right)$$

 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle =2xy^{\frac{3}{2}}+\frac{3}{x}+\left(\frac{3}{2}x^2y^{\frac{1}{2}}+\frac{2}{y}\right)y'$$
 *
$$\displaystyle (Eq.2.7.1)
 *
$$\displaystyle (Eq.2.7.1)
 * }
 * }

Let

 * $$\displaystyle G(y',y,x)=\frac{d}{dx}\phi(x,y)=0$$ .

Let

 * $$\displaystyle M(x,y)=2xy^{\frac{3}{2}}+\frac{3}{x}$$ ;

 * $$\displaystyle N(x,y)=\frac{3}{2}x^2y^{\frac{1}{2}}+\frac{2}{y} $$ .

Therefore, $$\displaystyle (Eq.2.7.1)$$ can be shown as

 * $$\displaystyle M(x,y)+N(x,y)y'=0$$ ,

which is the particular class of N1-ODE (linear in $$\displaystyle y'$$).
 Solution by EGM6321.f11.team1.pan

= R*2.8 First Exactness Condition =

Given

 * {| style="width:100%" border="0"

$$M\left(x,y\right)\cos y'+N(x,y)\log y'=0$$
 *
$$\,(Equation\;2.8.1) $$
 * style= |
 * }

== Find ==

Does Equation 2.8.1 satisfy the first exactness condition?

Basic Equations
First exactness condition for N1-ODE

 * $$M\left(x,y\right)+N(x,y)y'=0$$

Solutions
Let us define

$$\bar M\left(x,y\right):=M(x,y)$$

$$\bar N\left(x,y\right):=N(x,y)$$

Then Equation 2.8.1 becomes

$$\bar M\left(x,y\right)\cos y'+\bar N\left(x,y\right)\log y'=0$$

$$\bar M\left(x,y\right)+\bar N\left(x,y\right)\frac{\log y'}{\cos y'}=0$$

$$\bar M\left(x,y\right)+\bar N\left(x,y\right)F(y')=0$$

According to notes, if the function $$F$$ has no explicit inverse, then it CANNOT be put in the form which satisfies the First Exactness Condition.

The function $$F(y')=\frac{\log y'}{\cos y'}$$ has no inverse $$F^{-1}\left(y'\right)$$, and so therefore this N1-ODE fails to satisfy the First Exactness Condition.

 Solution by Egm6321.f11.team1.colsonfe 04:30, 17 September 2011 (UTC)

= R*2.9 State and proof the full theorem =

Given
Review calculus, and find the minimum degree of differentiability of the function such that (2) p.9-3 is satisfied. State the full theorem and provide a proof. p.9-3 (2): $$ \frac{\partial^2 \phi(x,y)}{\partial x \partial y}=\frac{\partial^2 \phi(x,y)}{\partial y \partial x} $$

== Find ==

See reference for full problem statement.

Clairaut's theorem
In mathematical analysis, Clairaut's theorem or Schwarz's theorem,[1] named after Alexis Clairaut and Hermann Schwarz, states that if $$f:\mathbb R \rightarrow \mathbb R$$ has continuous second partial derivatives at any given point in $$ \mathbb R^n $$,say $$(a_1,...,a_n)$$, then for $$ 1<i,$$  $$ j<n,$$

$$\frac{\partial^2 f}{\partial x_i \partial x_j}(a_1,...,a_n)=\frac{\partial^2 f}{\partial x_j \partial x_i}(a_1,...,a_n)$$

In words, the partial derivations of this function are commutative at that point. One easy way to establish this theorem (in the case where n = 2, i = 1, and j = 2, which readily entails the result in general) is by applying Green's theorem to the gradient of f.

http://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Clairaut.27s_theorem

Solution
The minimum degree of differentiability of the function such that (2) p.9-3 is satisfied is 3.

As Clairaut's theorem mentions, functions that satisfy (2) p.9-3 must have continuous second derivative.

For instance,

$$ z=f(x,y)=x^2+y^2 $$

$$\frac{\partial^2 f(x,y)}{\partial x\partial y}=\frac{\partial^2 f(x,y)}{\partial y\partial x}=2$$

The function

$$ z=f(x,y)=x^2+y^2 $$

can only be differentialed by 3 degrees

$$ \frac{\partial^2 f(x,y)}{\partial^2 x}=0 $$

And functions with degree of differentiability less than 3 can not satisfy (2) p. 9-3

For example f(x,y)=x+y

So The minimum degree of differentiability of the function such that (2) p.9-3 is satisfied is 3.

= R*2.10 Verify the solution for the N1-ODE =

Given

 * {| style="width:100%" border="0"

$$  \displaystyle y(x)=\sin ^{-1}(k-15x^5) $$     (10-1) (1) p.10-4
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \frac{d\phi (x,y(x))}{dx}=75x^4+(\cos y)y'=0 $$     (10-2) (2) p.8-6
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

== Find == Verify that 10-1 is the solution for the N1-ODE 10-2

Solution
We just need to put 10-1 into 10-2 and see if the equation is still in effect.
 * {| style="width:100%" border="0"

$$  \displaystyle \frac{d\phi (x,y(x))}{dx}=75x^4+(\cos (\sin ^{-1}(k-15x^5)))(\sin^{-1}(k-15x^{5}))' $$     (10-3) We can easily verify that the equation 10-3 is equal to 0 with the help of Matlab. i.e.
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$  \displaystyle 75x^4+(\cos (\sin ^{-1}(k-15x^5)))(\sin^{-1}(k-15x^{5}))'=0 $$     (10-4) Here are the codes:
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

syms x syms k syms y y=asin(k-15*x^5); dy=diff(y,x); 75*x^4+cos(y)*dy

This problem is solved by ourselves

= R2.11 Explain Difficulty In Solving For Integrating Factor = Question from 11-1: Why is it usually difficult to solve for the integrating factor h(x,y) in the following equation? $$h_xN - h_yM + h(N_x - M_y) = 0$$ The solution requires an understanding of how the Euler Integrating Factor Method (IFM) is derived with the reduction of order method. The solution expands upon concepts introduced in meetings 7-11.

First, the general nonlinear ODE of order n without the missing function y(x) is:

$$G(y^{(n)},y^{(n-1)},...,y'',y',y,x) = 0 $$

If we remove y, then the general nonlinear ODE without the function is:

$$G(y^{(n)},y^{(n-1)},...,y'',y',y,x) = 0 $$

Reduce the order by substituting p(x) := y'(x) below

$$G(p^{(n-1)},p^{(n-2)},...,p',p,x) = 0$$

The substitution gives the following solution for y(x).

$$y(x) = \int p(x)dx + k$$

where k is the integration constant. The key is in the dummy integration variable derivation below.

$$\int_a^x p(z)dz = \int_a^x y'(z)dz = y(x) - y(a)$$

Rearrangment gives

$$y(x) = \int_a^x p(z)dz + y(a) = P(x) - P(a) + y(a) = P(x) + k$$

The derivation requires the astute reader to note the distinction beween capital "P" and small "p." In addition, k = -P(a) + y(a)

Now, the Euler IFM applies to G(y',y,x)=0, and for a particular class of N1 ordinary differential equations, a solution linear in y' is

M(x,y) + N(x,y)y' = 0 where

$$ y' = \frac{dy}{dx}$$

So a particular solution is M(x,y) + N(x,y)dy = 0

The derivations to this point are from meeting 7 A more general solution for f(y') of the particular case is

$$\bar{M}(x,y) + \bar{N}(x,y)F(y') = 0$$

for the nonlinear case. But the assumption is that F(*) has an inverse. This leads to the first exactness conditions of N1 ordinary differential equations.

First Exactness Condition of N1-ODEs

If an N1-ODE is exact then it has a particular form of

M(x,y) + N(x,y)y' = 0 where

$$ y' = \frac{dy}{dx}$$

The appended derivations to this point are from meeting 8 Second Exactness Condition of N1-ODEs

Now, we must derive the 2nd exactness condition. A general ND-ODE is exact if for G the following relations exist:

$$\frac{d\phi}{dx} = G$$

$$\phi(x,y) = k$$

$$\frac{d\phi(x,y(x))}{dx} = G(y',y,x) = 0$$

Let $$\phi(x,y)$$ be a smooth function, then for the 2nd exactness condition we have

$$M_y(x,y) = \frac {\partial^2\phi(x,y)}{\partial x\partial y} = N_x(x,y) = \frac{\partial^2\phi(x,y)}{\partial y\partial x}$$
The appended derivations to this point are from meeting 9 If an N1-ODE satisfies the 1st exactness condition but not the 2nd, then the N1-ODE can be transformed by using the Euler Integrating Factor Method (IFM). Assume the conditions below.

$$M(x,y) + N(x,y)y' = 0 \And M_y(x,y) \ne N_x(x,y)$$

The transformation of the given N1-ODE to a form that fulfills both exactness conditions begins by finding an integrating factor h(x,y).

For condition 1, the substitution of h gives

$$(hM) + (hN)y' = \bar{M} + \bar{N}y' = 0$$

The key is to derive a solution using

$$\bar{M} = hM \And \bar{N} = hN$$

The derivations are as follows:

$$\bar{M}_y = h_yM + hM_y = \bar{N}_x = h_xN + hN_x$$

So the Euler IFM gives the following solution using h(x,y) as the transformation.

$$h_xN - h_yM + h(N_x - M_y) = 0 $$

The cumulative derivations to this point are from meetings 10-11 The solution h(x,y) is difficult to find since a separation of variables may not be possible for x and y. It is difficult to solve for h(x,y) in the equation when N(x,y) and M(x,y) are multivariate.

In addition, the following variables may have non-zero values, which complicates integrations to solve for h. The non-zero conditions complicate integration for h

$$h_x \ne 0$$ and $$h_y \ne 0$$

$$N_x \ne 0$$ and $$M_y \ne 0$$

So the solution of h is often cumbersome to solve because of the multivariate dependencies of h(x,y), M(x,y), and N(x,y) in the following equation.

$$\frac {h_x}{h} = -\frac{1}{N}(N_x - M_y)$$

The integration of this equation above is difficult with dependencies on both x and y and the non-zero conditions listed.

References for R2.11 Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL, (Meeting 7) 5 Sep 2011.

Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL, (Meeting 8) 6 Sep 2011.

Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL, (Meeting 9) 8 Sep 2011.

Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL, (Meeting 10) 8 Sep 2011.

Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL, (Meeting 11) 13 Sep 2011.

= R*2.12 Find Integrating Factor = Question from 11-3 : Find h for the following equation where h is the Euler Integration Factor Method. $$\frac{h_y}{h} = \frac{1}{M}(N_x - M_y) =: m(y)$$

The solution provides a condition in which simplification allows easier integration for one variable y

$$h_x(x,y) = 0$$

The condition above simplifies the integral for the Euler Integration Factor Method and the transformation involving h(x,y).

$$h_xN - h_yM + h(N_x - M_y) = -h_y + h(N_x - M_y) = 0$$

Rearrangement gives

$$h_yM = h(N_x - M_y) = 0$$

$$\frac{h_y}{h} = \frac{1}{M}(N_x - M_y) =: m(y)$$

$$\int \frac{h_y}{h} = - \int \frac {1}{N}(N_x - M_y)$$

Substitution of m(s) for the right side of the equation gives

$$ \log h(y) = \int\limits^{y} m(s)ds + k$$

So the solution is

$$h(y) = \exp {[\int\limits^{y} m(s)ds + k]}$$

The derivations used notes from meeting 11 References for R2.12 Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL, (Meeting 11) 13 Sep 2011.

= R*2.13 Prove Given Equation = Question from 11-5 : If the following equation is given $$y' + \frac{1}{x} = x^2$$

then show that

$$h(x) = x \And y(x) = \frac{x^3}{4} + \frac{k}{x}$$

The equation does not fulfill the 2nd exactness condition, so an integration factor h can be derived by rearranging the equation.

$$ a_0(x)y + y' = M(x,y) + N(x,y)y' = b(x)$$

Recall that h is defined as follows:

$$If h_y(x,y) = 0 then$$

$$\frac{h_x}{h} = \frac{1}{M}(N_x - M_y) =: n(x) = +a_0(x)$$

This gives h(x) as

$$h(x) = \exp[\int\limits^{x} a_0(s)ds + k_1]$$

Recall that

$$a_0(x) = \frac{1}{x}$$

So that $$h(x) = \exp[\int\limits^{x} \frac{1}{s}] = \exp(\ln {x}) = x $$

Now that we have solved for h(x), the next step is to solve for y(x). The transformation with h(x) is done here.

$$h(y' + a_0y) = hb$$

$$hy' + h'y = hb = (hy)'$$

Integration (hy)' to get

$$\int\limits^{x}(hy)' = \int\limits^{x}h(s)b(s)ds + k_2$$

Divide both sides by h(x) to get y(x).

$$y(x) = \frac{1}{h(x)}[\int\limits^{x}h(s)b(s)ds + k_2]$$

Recall that the given equation gives b as shown below.

$$h(x) = x \And b(x) = x^2$$

So substitute the functions back into the integral for y(x).

$$y(x) = \frac{1}{x}[\int\limits^{x} x^2 ] = \frac{1}{x}[\frac{x^4}{4} + k]$$ Simplification gives the solution for y(x) below.

$$y(x) = \frac{x^3}{4} + \frac{k}{x}$$

References for R2.13 Vu-Quoc, L. Class Lecture: Principles of Engineering Analysis. University of Florida, Gainesville, FL, (Meeting 11) 13 Sep 2011.

= R*2.14 Solve the general L1-ODE-VC=

== Part 1 == Solve the general L1-ODE-VC

Given
$$ a_1(x)\,y'+a_0(x)\,y=b(x)$$
$$\displaystyle (Equation\;14.1) $$

where

$$ a_1(x)=1,

a_0(x)=x,

b(x)=2x+3 $$

Solution
Substituting the conditions into 14.1

$$ y'+xy=2x+3$$
$$\displaystyle (Equation\;14.2) $$

Equation 14.2 is of the form

$$ y'+P(x)y=Q(x)$$

The integrating factor for this form is

$$ I(x)=\exp[P(x)dx]$$

For equation 14.2, the integrating factor is

$$ I(x)=\exp[\int xdx]=\exp[\frac{x^2}{2}] $$

Applying the integrating factor for equation 14.2:

$$ \exp[\frac{x^2}{2}]\frac{dy}{dx}+\exp[\frac{x^2}{2}]xy=2x\exp[\frac{x^2}{2}]+3\exp[\frac{x^2}{2}] $$
$$\displaystyle (Equation\;14.3) $$

(Eq.14.3) reduces to

$$ \frac{d}{dx}(y\exp(\frac{x^2}{2}))=2x\exp[\frac{x^2}{2}]+3\exp[\frac{x^2}{2}] $$

Integrating

$$ y\exp(\frac{x^2}{2})=\int 2x\exp[\frac{x^2}{2}]dx+\int3\exp[\frac{x^2}{2}]dx $$

Results in the following equation:

$$ y\exp(\frac{x^2}{2})=2\exp[\frac{x^2}{2}]+\int 3\exp [\frac{x^2}{2}]dx $$

There is no known anti-derivative of the last term, so using mathematical analysis tools the last therm results in the error-function. So that rearranging the equation we get for y(x):

$$ y=2+3\exp[\frac{x^2}{2}][\sqrt{\frac{\pi}{2}}erfi(\frac{x}{\sqrt2})] $$

==Part 2 == A general solution from the integrating factor method

Given
Assume $$ a(x)\ne0 $$ for all x, rearranging equationg 14.1 gives $$ y'+\frac{a_0(x)}{a_1(x)}y=\frac{b(x)}{a_1(x)} $$
$$\displaystyle (Equation\;14.4) $$

Find
Find an expression for $$ y(x) $$ in terms of $$ a_0, a_1 $$ and b

Solution
Define $$P(x)=\frac{a_0(x)}{a_1(x)}$$ and $$Q(x)=\frac{b(x)}{a_1(x)}$$

Equation 14.4 is then in the form

$$y'+P(x)y=Q(x)$$

The integrating factor for this form is

$$I(x)=\exp[\int{P(x)}dx]$$

Introducing a dummy factor t and substituting in for P(x)

$$\exp[\int^x{\frac{a_0(t)}{a_1(t)}}dt]\frac{dy}{dx}+\exp[\int^x{\frac{a_0(t)}{a_1(t)}}dt]\frac{a_0(x)}{a_1(x)}y=\frac{b(x)}{a_1(x)}\exp[\int^x{\frac{a_0(t)}{a_1(t)}}dt] $$

Reducing

$$\frac{d}{dx}[\exp[\int^x\frac{a_0(t)}{a_1(t)}]y]=\frac{b(x)}{a_1(x)}\exp[\int^x\frac{a_0(t)}{a_1(t)}dt]$$

Integrating

$$y=A\exp[-\int\frac{a_0(t)}{a_1(t)}dt]+\exp[-\int^x\frac{a_0(t)}{a_1(t)}dt]\int^x\frac{b(s)}{a_1(s)}ds\exp[-\int^s\frac{a_0(t)}{a_1(t)}dt]ds$$

==Part 3 == Solution to linear, first order, ordinary differential equation with varying coefficients using Euler integrating factor method.

Solution
Given an equation in the form of 14.1, where

$$a_1(x)=x^2+1, a_0(x)=x,$$ and $$b(x)=2x,$$

Equation 14.1 becomes

$$(x^2+1)y'+xy=2x$$

Rearranging

$$y'+\frac{x}{x^2+1}y=\frac{2x}{x^2+1}$$
$$\displaystyle (Equation\;14.5) $$

Equation 14.5 is in the form

$$y'+P(x)y=Q(x)$$||
$$\displaystyle (Equation\;14.6) $$

The integrating for this form is

$$I(x)=\exp[\int{\frac{x}{x^2+1}}dx]=\exp[\frac{1}{2}\ln(x^2+1)]=\sqrt {x^2+1} $$
$$\displaystyle (Equation\;14.7) $$

Multiply 14.6 by 14.7 and simplify

$$\frac{d}{dx}(y\sqrt {x^2+1})=\frac{2y}{\sqrt {x^2+1}}$$
$$\displaystyle (Equation\;14.8) $$

Integrating

$$\int^x\frac{d}{dx}(y\sqrt {x^2+1})=\int^x \frac{2y}{\sqrt {x^2+1}}dy$$
$$\displaystyle (Equation\;14.9) $$

Gives

$$Y=2$$

=R*2.15 Show the necessity of constant k1 and k2=

Given
Since (2)-(3) p.11-3 is an L1-ODE-VC, there should be only one integration constant, not two.

k1 is contained in (3) p11.4

$$ h(x)= \exp[ \int a_0(s)ds +k_1]$$

k2 is contained in (1) p11.5

$$y(x)=\frac{1}{h(x)}[\int^xh(x)b(s)ds+k_2]$$

== Find ==

Show that the integration constant k1 in (3 ) p.11-4 is not necessary, i.e, only k2 in (1) p.11-5 is necessary.

Solution
To show this we will use (6)p10.3 and show that k1 cancels out of the equation, but we must find the contants k1 and k2 from equations(eq8.1) and (eq8.2)

$$y(x)=\frac{1}{h(x)}\int^xh(s)b(s)ds$$

When we perform the integration in the RHS of (eq8.1) it becomes

$$\exp[\int^x a_0(s)ds]=\exp[\int a_0(x)dx+k]$$

Using the properties for exponential functions

$$\exp[a+b]=\exp[a]exp[b]$$

Now (eq8.3) becomes

$$\exp[\int a_0(x)dx]\underbrace{\exp[k]}_{=:k_1}$$

Plugging this back into (eq8.1) gives us

$$h(x)= k_1\exp[\int a_0(x)dx]$$

To find k2 we will evaluate the integral in (eq8.2)

$$\int^x h(s)b(s)ds=\int h(x)b(x)dx+k_2$$

Finally plug equations (eq8.5) and (eq8.6) into (6) p.10-3

$$y(x)=\frac{1}{\cancelto{}{k_1}\exp[\int a_0(x)dx]}[\int \cancelto{}{k_1}\exp[\int a_0(x)dx]b(x)dx+k_2]$$

This shows k1 is not necessary

$$y(x)=\frac{1}{\exp[\int a_0(x)dx]}[\int \exp[\int a_0(x)dx]b(x)dx+k_2]$$

=R*2.16 LVQ Solution compared to King =

Given
The solution of

$$\displaystyle a_1(x) \cdot y^{(1)}+a_0(x)y^{(0)}=b(x)$$
$$\displaystyle (3) p. 11-3 $$

is

$$\displaystyle y(x) = \frac{1}{h(x)} \left[\int^x h(s)b(s)ds + k_2 \right] $$
$$\displaystyle (1) p. 11-5 $$

From lecture notes [[media:pea1.f11.mtg11.djvu|Mtg 11 (b)]]

== Find ==

Show that $$\displaystyle (1) P. 11-5 $$ agrees with $$\displaystyle King p. 512 $$ shown below

 * {| style="width:100%" border="0"

y = A{y}_{H}\left(x \right)+{y}_{P}\left(x \right) $$ $$
 *
$$\displaystyle (2.16.1)
 * style= |
 * }

Solution
Start by using the integration factor defined by $$\displaystyle (3) p. 11-4$$

$$ h(x) = \exp \left[ \int^{x} {a}_{0} \left( s\right)ds + k_1 \right]$$
$$\displaystyle (3) p. 11-4 $$

Using the properties for exponential functions $$ \exp \left[ a+b\right] = \exp \left[ a\right]\exp \left[b\right]$$

Now $$\displaystyle (3) p. 11-4$$ becomes

 * {| style="width:100%" border="0"

\exp \left[ \int {a}_{0} \left( x\right)dx \right]\underbrace{\exp \left[k\right]}_{=: k_1} $$ $$
 *
$$\displaystyle (2.16.2)
 * style= |
 * }

Plugging this back into $$\displaystyle (3) p. 11-4 $$ gives us

$$ h \left( x\right)= k_{1}\exp \left[ \int {a}_{0} \left( x\right)dx\right] $$
$$\displaystyle (Eq 2.16.3) $$

To find $$ {k}_{2}$$ we will evaluate the integral in $$\displaystyle (1) p. 11-5$$

$$\int^{x} h\left( s\right)b\left( s\right)ds= \int h\left( x\right)b\left( x\right)dx + k_{2}$$
$$\displaystyle (Eq 2.16.4) $$

Next plug equations $$\displaystyle Eq 2.16.3 $$ and $$\displaystyle Eq 2.16.4 $$ into $$\displaystyle (1) p. 11-5 $$ and canceling $$\displaystyle k_1 $$

$$y \left( x\right)= \frac{1}{\exp \left[  \int {a}_{0} \left( x\right)dx\right]}\left[ \int \exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx +k_2\right]$$
$$\displaystyle (Eq 2.16.5) $$

From $$\displaystyle K p. 512 $$ the equation for the solution of the homogeneous differential equation and the particular integral solution of the inhomogeneous differential equation are given below, respectively

 * {| style="width:100%" border="0"

A{y}_{H}\left(x \right)= A\exp \left\{ - \int ^{x}{P} \left( t\right)dt\right\} $$ $$
 *
$$\displaystyle (Eq 2.16.6)
 * style= |
 * }

 * {| style="width:100%" border="0"

{y}_{P}\left(x \right)= \exp \left\{ - \int ^{x}{P} \left( t\right)dt\right\}\int ^{x}{Q} \left( s\right)\exp \left\{  \int ^{x}{P} \left(t\right)dt\right\}ds $$ $$
 *
$$\displaystyle (Eq 2.16.7)
 * style= |
 * }

Now, using $$\displaystyle (Eq 2.16.5) $$ which is the expanded form of $$\displaystyle (3) p. 11-4 $$ we can further expand and rearrange to get

 * {| style="width:100%" border="0"

y \left( x\right)= \frac{1}{\exp \left[  \int {a}_{0} \left( x\right)dx\right]} \int \exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx +\frac{k_2}{\exp \left[  \int {a}_{0} \left( x\right)dx\right]} $$ $$
 *
$$\displaystyle (Eq 2.16.8)
 * style= |
 * }

 * {| style="width:100%" border="0"

y \left( x\right)= \exp \left[  -\int {a}_{0} \left( x\right)dx\right] \int \exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx +k_2\exp \left[  -\int {a}_{0} \left( x\right)dx\right] $$ $$
 *
$$\displaystyle (Eq 2.16.9)
 * style= |
 * }

Now we have $$\displaystyle (3) p. 11-4 $$ in the form of $$\displaystyle (Eq 2.16.1)$$ where

 * {| style="width:100%" border="0"

A{y}_{H}\left(x \right)= k_2\exp \left[ -\int {a}_{0} \left( x\right)dx\right] $$ $$
 *
$$\displaystyle (Eq 2.16.10)
 * style= |
 * }

 * {| style="width:100%" border="0"

{y}_{P}\left(x \right)= \exp \left[ -\int {a}_{0} \left( x\right)dx\right] \int \exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx $$ $$
 *
$$\displaystyle (Eq 2.16.11)
 * style= |
 * }

By comparing $$\displaystyle (Eq 2.16.6) $$ with $$\displaystyle (Eq 2.16.10)$$ and $$\displaystyle (Eq 2.16.7) $$ with $$\displaystyle (Eq 2.16.11) $$ we can see that these equations are in agreement

= R*2.17 Redo the equation for the solution of the homogeneous differential equation done R*2.16 using (2) p. 12-2 =

Given

 * {| style="width:100%" border="0"
 * {| style="width:100%" border="0"

$$ From lecture notes [[media:pea1.f11.mtg11.djvu|Mtg 11 (b)]]
 * $$\displaystyle {{y}^{'}}+{{a}_{0}(x)}y=0$$
 *
$$\displaystyle (2) p. 12-2
 *
$$\displaystyle (2) p. 12-2
 * }
 * }

== Find ==
 * {| style="width:100%" border="0"

 * Find $$\displaystyle y_H$$
 * style= |
 * }
 * }

Solution
Since $$\displaystyle (2) p. 12-2 $$ is the homogeneous portion of

$$\displaystyle a_1(x) \cdot y^{(1)}+a_0(x)y^{(0)}=b(x)$$ with $$\displaystyle  a_1(x)=1$$
$$\displaystyle (3) p. 11-3 $$

Equation $$\displaystyle (2) p. 12-2 $$  is seperable, So we can solve directly for $$ y_H$$

Equation $$\displaystyle (2) p. 12-2 $$ can be rearranged as follows

 $$\displaystyle \frac{y'}{y} = -{a}_{0} $$
$$\displaystyle (Eq 2.17.1) $$

Integration of both sides of this equation gives

 $$ \ln \left( y\right)= -\int^{x} {a}_{0} \left( s\right)ds + C$$
$$\displaystyle (Eq 2.17.2)$$

Solving for $$y$$ gives

$$ y= \exp\left[ -\int^{x} {a}_{0} \left( s\right)ds + C \right]= A\exp\left[ -\int^{x} {a}_{0} \left( s\right)ds\right]= y_H $$
$$\displaystyle (Eq 2.17.3) $$

= R*2.18 - Find the integrating factors to make L1-ODE-VC exact =

Given
$$\displaystyle \left[ x^4 y + 10 \right] + \frac{1}{2} x^2 y' = 0 $$
$$\displaystyle (4) p. 12.4 $$

From lecture notes [[media:pea1.f11.mtg12.djvu|Mtg 12]]

== Find ==

Determine if $$\displaystyle (4) p. 12-4 $$ is exact, if it is not find the integrating factor $$\displaystyle h $$ to make it exact

Solution
The first exactness condition for an N1-ODE is

$$\displaystyle M(x,y) + N(x,y) y'= 0$$
$$\displaystyle (2) p. 7-6 $$

Since the equation is already in this form the first exactness condition is fulfilled

The second exactness condition requires that

$$\displaystyle M_y(x,y)=N_y(x,y)$$
$$\displaystyle (3) p. 9-3 $$

Where:

$$\displaystyle M(x,y) = \left[x^4y+10\right]$$
$$\displaystyle (2.18.1) $$

And:

$$\displaystyle N(x,y) = \left (\frac{1}{2}x^2\right)$$
$$\displaystyle (2.18.2) $$

Taking the derivatives:

$$\displaystyle M_y(x,y)=x^4$$
$$\displaystyle (2.18.3) $$

$$\displaystyle N_x(x,y)= x$$
$$\displaystyle (2.18.4) $$

Since $$\displaystyle M_y(x,y) \ \ne N_y(x,y) $$
$$\displaystyle (2.18.5) $$

Therefore equation $$\displaystyle (4) p. 12-4 $$ is not exact

So we re-write the ODE

 * $$ \underbrace{\frac{1}{2}x^{2}}_{N(x,y)}y'+\underbrace{[x^{4}y]}_{M(x,y)}= -10 $$

Looking at the first exactness condition for the integrating factor we obtain:

$$\displaystyle \frac{1}{N}\left(N_x - M_y\right) = -f(x) $$
$$\displaystyle (2.18.6) $$ replacing values found above $$\displaystyle \frac{2}{x^2}\left(x - x^4\right) = -f(x) $$
$$\displaystyle (2.18.7) $$ applying knowledge from integrating factor we know $$\displaystyle h(x)=\exp \int^{x}f(s)ds $$
$$\displaystyle (2.18.8) $$ replacing f(x) into the integral and evaluating it, obtain $$\displaystyle h(x) = \frac{1}{x^2}\exp \left(\frac{2x^3}{3}\right)$$
$$\displaystyle (2.18.9) $$

Lets verify that the integrating factor has made the ODE exact by checking the second condition of exactness $$\displaystyle \bar{M}_y = \bar{N}_x $$
$$\displaystyle (2.18.10) $$ $$\displaystyle\underbrace{(hM)}_{\bar{M}}dx = \underbrace{(hN)}_{\bar{N}}dy$$
$$\displaystyle (2.18.11) $$

Calculating the respective values we obtain

$$\displaystyle \bar{N}_x = x^2 \exp\left(\frac{2x^3}{3}\right)$$
$$\displaystyle (2.18.12) $$ $$\displaystyle \bar{M}_y =x^2 \exp\left(\frac{2x^3}{3}\right)$$
$$\displaystyle (2.18.13) $$

Since $$\displaystyle (2.18.10)$$ is satisfied(i.e. $$\displaystyle ((2.18.12) = (2.19.13))$$ the non-homogenous L1_ODE_VC has been made exact by using the appropriate integrating factor $$\displaystyle h $$.

= References =

= Table of Assignments =

{| class="prettytable" style="margin: 1em auto 1em auto;"
 * colspan="2" | Problem Assignments

 * Problem #
 * Solved by
 * Solved by

 * 1
 * Jing Pan
 * Jing Pan

 * 2
 * Jing Pan
 * Jing Pan

 * 3
 * Zexi Zheng
 * Zexi Zheng

 * 4
 * Zexi Zheng
 * Zexi Zheng

 * 5
 * Fenner Colson
 * Fenner Colson

 * 6
 * Fenner Colson
 * Fenner Colson

 * 7
 * Jing Pan
 * Jing Pan

 * 8
 * Fenner Colson
 * Fenner Colson

 * 9
 * Yi Zhao
 * Yi Zhao

 * 10
 * Zexi Zheng
 * Zexi Zheng

 * 11
 * Manuel Steele
 * Manuel Steele

 * 12
 * Manuel Steele
 * Manuel Steele

 * 13
 * Manuel Steele
 * Manuel Steele

 * 14
 * Yi Zhao
 * Yi Zhao

 * 15
 * Yi Zhao
 * Yi Zhao

 * 16
 * Ben Neri
 * Ben Neri

 * 17
 * Ben Neri
 * Ben Neri

 * 18
 * Ben Neri
 * Ben Neri

 * -}

title: User:Egm6321.f11.team1/report2


text:
Engineering Projects/Poppit/Howard Community College/Fall2011/502 1+1=11/Full Code

Link to project 1+1=11 team page.

Full Processing code:

<!--


text:
User:Bmart14

[Poppit Project]

[MC Escher Project]

[MC Escher Project2]

[MC Escher Project3]

title: User:Bmart14


text:
User:Koalex/escher

Link to project ABM team page.

Give instructor top 3 project choice list
Typically the instructor picks teams. Students send the instructor their top 3 choices. The project choices are typically those found at Category:General_Engineering_Projects. The instructor tries to put students in the project of their choice. Put your top three choices here after they are used to assign you to this project. Create a link to the project root and to where your team will be working.

Write problem statement
Look through the existing problem statements. Find something not yet done on the project. Create a problem statement for it. Think of a new subsystem that needs to be built for the project. The goal is something simple, possible, for zero dollars.

Assign Task1
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Compare actual work done to Task1
Rationalize differences

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Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

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Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

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Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:Koalex/escher


text:
User:Inmookchung27

This is my escher page

 * [Poppit]

 * [escher]

 * [escher2]

 * [escher3]

title: User:Inmookchung27


text:
User:Bmart14/enes100/escher

Link Back [Link back to team page]

Give instructor top 3 project choice list
We gave Scott Forester our top 3 Picks which was

-MC Escher

-Coanda

-Wind Tunnel

Write problem statement
Create a perpetual motion waterfall machine to replicates M.C Escher's Design

Assign Task1
We all decided to go home and do massive research on M.C Escher and how to attempt to make the waterfall. We will all converge about ideas on tuesday.

Compare actual work done to Task1
The weekends work was successful and now we are converging all of our ideas into one.

Week1 Narrative
A problem we had to figure out during our week was how to get the water to go to the second level of the perpetual motion machine.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
We all divided assignments and I plan on moving the team forward by drawing sketches on Autodesk Inventor for blueprints and designs for the builder, Mike Chung

Compare actual work done to Task2
I drew up all of the models on Autodesk Inventor and emailed the drawings to Mike Chung, who is now building the Escher waterfall. I will also help him read to blueprints and build the waterfall.

Week2 Narrative
I am now continuing to work on the actual waterfall with Mike Chung and i am also re-sketching the autodesk sketches on 123D for more practice and fixing some bugs with my other design.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and Bmart14 16:37, 22 September 2011 (UTC) create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
I will finish up all of the Autodesk 123D drawings, along with sharing it with Mike Chung, and also take/upload pictures to wiki for further documention.

Compare actual work done to Task3
Over the weekend I spoke to the builder on the team, mike, to make sure all the drawings are up to scale, and they seem to be a little off. So i decided to go to my high school to use software that is was more familiar with (Autodesk Professional 2011)and am still in the process of recreating the drawings.

Week3 Narrative
Problems I had were using Autodesk 123D and the little things from dimensioning certain objects to extruding without changing ratios. My quick fix for this was going five minuets up the street and using software that ive used for 4 years and i am very familiar with.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Over the weekend I started working on re-creating the Waterfall we built on AutoDesk Inventor 2011. I used pictures and dimensions i took from my other teammate to draw the waterfall as accurately as i could. I am still currently working on the drawings on AutoDesk.

Week4 Narrative
My drawings on Autodesk Inventor are almost completed for the the re-creation of the waterfall. I still need to complete the 3rd level along with assembling all of the parts to mate together.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:Bmart14/enes100/escher


text:
User:Inmookchung27/enes100/escher

Link Back [Link back to team page]

Give instructor top 3 project choice list
escher

Coanda

Wind tunnel

Write problem statement
Look through the existing problem statements. Find something not yet done on the project. Create a problem statement for it. Think of a new subsystem that needs to be built for the project. The goal is something simple, possible, for zero dollars.

Assign Task1
Tasks are typically done over the weekend. They are documented during the week, presented during the week and the next weekends work is negotiated during the week. Week 0 is the week that task 1 is negotiated. Record your individual task here. Plan on working on it during the weekend between week0 and week1.

Compare actual work done to Task1
Rationalize differences

Week1 Narrative
The problem was how we are going to pump the water up from the bottom level of the escher recreation of the endless waterfall to the top level.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
i will move the team forward by informing my teammate that was not here what we are doing. I am gathering the material needed to build the endless waterfall, start building the waterfall and bryan martinelli is drawing the sketch of it out on autodesk

Compare actual work done to Task2
My partner bryan martinelli started to draw out the design on autodesk over the weekend, and he has sent me a copy of it so i started to build the Escher waterfall out of wood and other materials.

Week2 Narrative
I edited the team page and also bought the material to build the escher waterfall. I started to make the measurements.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Finish measuring out the material, and start putting together the pieces in order to create the waterfall. I will upload pictures on wikidoc with my partner bryan martinelli

Compare actual work done to Task3
I built the first floor and second floor, still have some more pictures to upload

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them. Some of the problems i had was deciding which glue to use to hold the wood together.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Compare actual work done to Task4
Over the weekend I put together the last level of the waterfall, and actually started to pour water on the waterfall just to see how it would hold up

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

title: User:Inmookchung27/enes100/escher


text:
User:Koalex/enes100/escher

Give instructor top 3 project choice list
Typically the instructor picks teams. Students send the instructor their top 3 choices. The project choices are typically those found at Category:General_Engineering_Projects. The instructor tries to put students in the project of their choice. Put your top three choices here after they are used to assign you to this project. Create a link to the project root and to where your team will be working.

Write problem statement
Look through the existing problem statements. Find something not yet done on the project. Create a problem statement for it. Think of a new subsystem that needs to be built for the project. The goal is something simple, possible, for zero dollars.

Assign Task1
Tasks are typically done over the weekend. They are documented during the week, presented during the week and the next weekends work is negotiated during the week. Week 0 is the week that task 1 is negotiated. Record your individual task here. Plan on working on it during the weekend between week0 and week1.

Compare actual work done to Task1
Rationalize differences

Week1 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
Record what you are planning on doing for the team during the weekend between week1 and week2 of the project here.

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

title: User:Koalex/enes100/escher


text:
Engineering Projects/Poppit/Howard Community College/Fall2011/502 ZTRH

Problem Statement
Put the last negotiated problem statement with the instructor or client of the project here.

Team Members
Put each team members name here with links to their personal page here. Real names do not have to be used.
 * Zach Thompson

Summary
Put an overall, one paragraph summary here with links to the team weekly reports.

Poster
Put a graphic in wikimedia, include the graphic here or link to it here. The graphic should be suitable for creating a traditional project poster.

Story
Tell a story of the project. Describe how split up, what the obstacles were, what testing was done, what informal decisions were made, what assumptions were made, what the results were. This is a longer version of the summary with links to all the details collected associated with the project. These links could be to software, links to videos, links to project pages with pictures, etc. Think of the story as a summary of the team weekly reports on one long page rather than a short paragraph like the summary.

Decision List
List all formal decisions made with links to their documentation such as a decision tree or decision matrix.

Material List
Figuring out what to purchase, what to build, what everything costs is a huge part of engineering. Typically there is a list of materials in stock, materials that are ordered, materials that should be ordered next time there is money, materials that have not been fully justified. These issues are part of healthy management of the engineering lab but are associated with a particular college. The detail needs to be published in this "Done" form. However in the project root, just a list of materials is necessary.

Software List
Installing and learning different software packages is part of most engineering project just like the materials list.

Time
Time estimates and actual time consumed measurements helps justify salaries (grades). The only way to gain respect for estimated project time and costs is to practice.

Tutorials
Most projects consist of making instructions to jump start the next team, shrink the learning of tools and software to the minimum, advice on where to purchase materials, how to assemble, etc. Tutorials modified or created are described here with links.

Next Steps
If this is not filled out, the next team may have to repeat each of your steps just to figure out what the next step is. It is best to write this immediately at the end of the project while creating the summary, poster, presentation, and story.