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ExplorerGuide

Course: CSCI 5523, Fall 2008
School: Minnesota
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Explorer WEKA User Guide for Version 3-4 Richard Kirkby Eibe Frank June 1, 2007 c 2002-2007 University of Waikato Contents 1 Launching WEKA 2 The WEKA Explorer Section Tabs . . . . . . Status Box . . . . . . . Log Button . . . . . . . WEKA Status Icon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

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Explorer WEKA User Guide for Version 3-4 Richard Kirkby Eibe Frank June 1, 2007 c 2002-2007 University of Waikato Contents 1 Launching WEKA 2 The WEKA Explorer Section Tabs . . . . . . Status Box . . . . . . . Log Button . . . . . . . WEKA Status Icon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 3 3 3 3 3 4 4 5 6 6 6 7 7 7 8 9 9 9 9 10 3 Preprocessing Opening les . . . . . . . The Current Relation . . Working With Attributes Working With Filters . . 4 Classication Selecting a Classier . . . . Test Options . . . . . . . . The Class Attribute . . . . Training a Classier . . . . The Classier Output Text The Result List . . . . . . . 5 Clustering Selecting a Clusterer Cluster Modes . . . Ignoring Attributes . Learning Clusters . . . . . . . . . . . . . . . . . . 6 Associating 10 Setting Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Learning Associations . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 7 Selecting Attributes 10 Searching and Evaluating . . . . . . . . . . . . . . . . . . . . . . . . . 10 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Performing Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 8 Visualizing 11 The scatter plot matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Selectin an individual 2D scatter plot . . . . . . . . . . . . . . . . . . 11 Selecting Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1 1 Launching WEKA The WEKA GUI Chooser window is used to launch WEKAs graphical environments. At the bottom of the window are four buttons: 1. Simple CLI. Provides a simple command-line interface that allows direct execution of WEKA commands for operating systems that do not provide their own command line interface. 2. Explorer. An environment for exploring data with WEKA. 3. Experimenter. An environment for performing experiments and conducting statistical tests between learning schemes. 4. Knowledge Flow. This environment supports essentially the same functions as the Explorer but with a drag-and-drop interface. One advantage is that it supports incremental learning. If you launch WEKA from a terminal window, some text begins scrolling in the terminal. Ignore this text unless something goes wrong, in which case it can help in tracking down the cause. This User Manual, which is also available online on the WekaDoc Wiki [3], focuses on using the Explorer but does not explain the individual data preprocessing tools and learning algorithms in WEKA. For more information on the various lters and learning methods in WEKA, see the book Data Mining [2]. 2 The WEKA Explorer Section Tabs At the very top of the window, just below the title bar, is a row of tabs. When the Explorer is rst started only the rst tab is active; the others are greyed out. This is because it is necessary to open (and potentially pre-process) a data set before starting to explore the data. The tabs are as follows: 1. Preprocess. Choose and modify the data being acted on. 2. Classify. Train and test learning schemes that classify or perform regression. 3. Cluster. Learn clusters for the data. 4. Associate. Learn association rules for the data. 5. Select attributes. Select the most relevant attributes in the data. 6. Visualize. View an interactive 2D plot of the data. Once the tabs are active, clicking on them icks between dierent screens, on which the respective actions can be performed. The bottom area of the window (including the status box, the log button, and the Weka bird) stays visible regardless of which section you are in. 2 Status Box The status box appears at the very bottom of the window. It displays messages that keep you informed about whats going on. For example, if the Explorer is busy loading a le, the status box will say that. TIPright-clicking the mouse anywhere inside the status box brings up a little menu. The menu gives two options: 1. Memory information. Display in the log box the amount of memory available to WEKA. 2. Run garbage collector. Force the Java garbage collector to search for memory that is no longer needed and free it up, allowing more memory for new tasks. Note that the garbage collector is constantly running as a background task anyway. Log Button Clicking on this button brings up a separate window containing a scrollable text eld. Each line of text is stamped with the time it was entered into the log. As you perform actions in WEKA, the log keeps a record of what has happened. WEKA Status Icon To the right of the status box is the WEKA status icon. When no processes are running, the bird sits down and takes a nap. The number beside the symbol gives the number of concurrent processes running. When the system is idle it is zero, but it increases as the number of processes increases. When any process is started, the bird gets up and starts moving around. If its standing but stops moving for a long time, its sick: something has gone wrong! In that case you should restart the WEKA explorer. 3 Preprocessing Opening les The rst three buttons at the top of the preprocess section enable you to load data into WEKA: 1. Open le.... Brings up a dialog box allowing you to browse for the data le on the local lesystem. 2. Open URL.... Asks for a Uniform Resource Locator address for where the data is stored. 3. Open DB.... Reads data from a database. (Note that to make this work you might have to edit the le in weka/experiment/DatabaseUtils.props.) Using the Open le... button you can read les in a variety of formats: Wekas ARFF format, CSV format, C4.5 format, or serialized Instances format. ARFF les typically have a .ar extension, CSV les a .csv extension, C4.5 les a .data and .names extension, and serialized Instances objects a .bsi extension. 3 The Current Relation Once some data has been loaded, the Preprocess panel shows a variety of information. The Current relation box (the current relation is the currently loaded data, which can be interpreted as a single relational table in database terminology) has three entries: 1. Relation. The name of the relation, as given in the le it was loaded from. Filters (described below) modify the name of a relation. 2. Instances. The number of instances (data points/records) in the data. 3. Attributes. The number of attributes (features) in the data. Working With Attributes Below the Current relation box is a box titled Attributes. There are three buttons, and beneath them is a list of the attributes in the current relation. The list has three columns: 1. No.. A number that identies the attribute in the order they are specied in the data le. 2. Selection tick boxes. These allow you select which attributes are present in the relation. 3. Name. The name of the attribute, as it was declared in the data le. When you click on dierent rows in the list of attributes, the elds change in the box to the right titled Selected attribute. This box displays the characteristics of the currently highlighted attribute in the list: 1. Name. The name of the attribute, the same as that given in the attribute list. 2. Type. The type of attribute, most commonly Nominal or Numeric. 3. Missing. The number (and percentage) of instances in the data for which this attribute is missing (unspecied). 4. Distinct. The number of dierent values that the data contains for this attribute. 5. Unique. The number (and percentage) of instances in the data having a value for this attribute that no other instances have. Below these statistics is a list showing more information about the values stored in this attribute, which dier depending on its type. If the attribute is nominal, the list consists of each possible value for the attribute along with the number of instances that have that value. If the attribute is numeric, the list gives four statistics describing the distribution of values in the datathe minimum, maximum, mean and standard deviation. And below these statistics there is a colored histogram, color-coded according to the attribute chosen as the Class using the box above the histogram. (This box will bring up a drop-down list 4 of available selections when clicked.) Note that only nominal Class attributes will result in a color-coding. Finally, after pressing the Visualize All button, histograms for all the attributes in the data are shown in a separate witting. Returning to the attribute list, to begin with all the tick boxes are unticked. They can be toggled on/o by clicking on them individually. The three buttons above can also be used to change the selection: 1. All. All boxes are ticked. 2. None. All boxes are cleared (unticked). 3. Invert. Boxes that are ticked become unticked and vice versa. Once the desired attributes have been selected, they can be removed by clicking the Remove button below the list of attributes. Note that this can be undone by clicking the Undo button, which is located next to the Edit button in the top-right corner of the Preprocess panel. Working With Filters The preprocess section allows lters to be dened that transform the data in various ways. The Filter box is used to set up the lters that are required. At the left of the Filter box is a Choose button. By clicking this button it is possible to select one of the lters in Weka. Once a lter has been selected, its name and options are shown in the eld next to the Choose button. Clicking on this box brings up a GenericObjectEditor dialog box. The GenericObjectEditor Dialog Box The GenericObjectEditor dialog box lets you congure a lter. The same kind of dialog box is used to congure other objects, such as classiers and clusterers (see below). The elds in the window reect the available options. Clicking on any of these gives an opportunity to alter the lters settings. For example, the setting may take a text string, in which case you type the string into the text eld provided. Or it may give a drop-down box listing several states to choose from. Or it may do something else, depending on the information required. Information on the options is provided in a tool tip if you let the mouse pointer hover of the corresponding eld. More information on the lter and its options can be obtained by clicking on the More button in the About panel at the top of the GenericObjectEditor window. Some objects display a brief description of what they do in an About box, along with a More button. Clicking on the More button brings up a window describing what the dierent options do. At the bottom of the GenericObjectEditor dialog are four buttons. The rst two, Open... and Save... allow object congurations to be stored for future use. The Cancel button backs out without remembering any changes that have been made. Once you are happy with the object and settings you have chosen, click OK to return to the main Explorer window. 5 Applying Filters Once you have selected and congured a lter, you can apply it to the data by pressing the Apply button at the right end of the Filter panel in the Preprocess panel. The Preprocess panel will then show the transformed data. The change can be undone by pressing the Undo button. You can also use the Edit... button to modify your data manually in a dataset editor. Finally, the Save... button at the top right of the Preprocess panel saves the current version of the relation in the same formats available for loading data, allowing it to be kept for future use. Note: Some of the lters behave dierently depending on whether a class attribute has been set or not (using the box above the histogram, which will bring up a drop-down list of possible selections when clicked). In particular, the supervised lters require a class attribute to be set, and some of the unsupervised attribute lters will skip the class attribute if one is set. Note that it is also possible to set Class to None, in which case no class is set. 4 Classication Selecting a Classier At the top of the classify section is the Classier box. This box has a text eld that gives the name of the currently selected classier, and its options. Clicking on the text box brings up a GenericObjectEditor dialog box, just the same as for lters, that you can use to congure the options of the current classier. The Choose button allows you to choose one of the classiers that are available in WEKA. Test Options The result of applying the chosen classier will be tested according to the options that are set by clicking in the Test options box. There are four test modes: 1. Use training set. The classier is evaluated on how well it predicts the class of the instances it was trained on. 2. Supplied test set. The classier is evaluated on how well it predicts the class of a set of instances loaded from a le. Clicking the Set... button brings up a dialog allowing you to choose the le to test on. 3. Cross-validation. The classier is evaluated by cross-validation, using the number of folds that are entered in the Folds text eld. 4. Percentage split. The classier is evaluated on how well it predicts a certain percentage of the data which is held out for testing. The amount of data held out depends on the value entered in the % eld. Note: No matter which evaluation method is used, the model that is output is always the one build from all the training data. Further testing options can be set by clicking on the More options... button: 6 1. Output model. The classication model on the full training set is output so that it can be viewed, visualized, etc. This option is selected by default. 2. Output per-class stats. The precision/recall and true/false statistics for each class are output. This option is also selected by default. 3. Output entropy evaluation measures. Entropy measures evaluation are included in the output. This option is not selected by default. 4. Output confusion matrix. The confusion matrix of the classiers predictions is included in the output. This option is selected by default. 5. Store predictions for visualization. The classiers predictions are remembered so that they can be visualized. This option is selected by default. 6. Output predictions. The predictions on the evaluation data are output. Note that in the case of a cross-validation the instance numbers do not correspond to the location in the data! 7. Cost-sensitive evaluation. The errors is evaluated with respect to a cost matrix. The Set... button allows you to specify the cost matrix used. 8. Random seed for xval / % Split. This species the random seed used when randomizing the data before it is divided up for evaluation purposes. The Class Attribute The classiers in WEKA are designed to be trained to predict a single class attribute, which is the target for prediction. Some classiers can only learn nominal classes; others can only learn numeric classes (regression problems); still others can learn both. By default, the class is taken to be the last attribute in the data. If you want to train a classier to predict a dierent attribute, click on the box below the Test options box to bring up a drop-down list of attributes to choose from. Training a Classier Once the classier, test options and class have all been set, the learning process is started by clicking on the Start button. While the classier is busy being trained, the little bird moves around. You can stop the training process at any time by clicking on the Stop button. When training is complete, several things happen. The Classier output area to the right of the display is lled with text describing the results of training and testing. A new entry appears in the Result list box. We look at the result list below; but rst we investigate the text that has been output. The Classier Output Text The text in the Classier output area has scroll bars allowing you to browse the results. Of course, you can also resize the Explorer window to get a larger display area. The output is split into several sections: 7 1. Run information. A list of information giving the learning scheme options, relation name, instances, attributes and test mode that were involved in the process. 2. Classier model (full training set). A textual representation of the classication model that was produced on the full training data. 3. The results of the chosen test mode are broken down thus: 4. Summary. A list of statistics summarizing how accurately the classier was able to predict the true class of the instances under the chosen test mode. 5. Detailed Accuracy By Class. A more detailed per-class break down of the classiers prediction accuracy. 6. Confusion Matrix. Shows how many instances have been assigned to each class. Elements show the number of test examples whose actual class is the row and whose predicted class is the column. The Result List After training several classiers, the result list will contain several entries. Leftclicking the entries icks back and forth between the various results that have been generated. Right-clicking an entry invokes a menu containing these items: 1. View in main window. Shows the output in the main window (just like left-clicking the entry). 2. View in separate window. Opens a new independent window for viewing the results. 3. Save result buer. Brings up a dialog allowing you to save a text le containing the textual output. 4. Load model. Loads a pre-trained model object from a binary le. 5. Save model. Saves a model object to a binary le. Objects are saved in Java serialized object form. 6. Re-evaluate model on current test set. Takes the model that has been built and tests its performance on the data set that has been specied with the Set.. button under the Supplied test set option. 7. Visualize classier errors. Brings up a visualization window that plots the results of classication. Correctly classied instances are represented by crosses, whereas incorrectly classied ones show up as squares. 8. Visualize tree or Visualize graph. Brings up a graphical representation of the structure of the classier model, if possible (i.e. for decision trees or Bayesian networks). The graph visualization option only appears if a Bayesian network classier has been built. In the tree visualizer, you can bring up a menu by right-clicking a blank area, pan around by dragging the mouse, and see the training instances at each node by clicking on it. CTRL-clicking zooms the view out, while SHIFT-dragging a box zooms the view in. The graph visualizer should be self-explanatory. 8 9. Visualize margin curve. Generates a plot illustrating the prediction margin. The margin is dened as the dierence between the probability predicted for the actual class and the highest probability predicted for the other classes. For example, boosting algorithms may achieve better performance on test data by increasing the margins on the training data. 10. Visualize threshold curve. Generates a plot illustrating the tradeos in prediction that are obtained by varying the threshold value between classes. For example, with the default threshold value of 0.5, the predicted probability of positive must be greater than 0.5 for the instance to be predicted as positive. The plot can be used to visualize the precision/recall tradeo, for ROC curve analysis (true positive rate vs false positive rate), and for other types of curves. 11. Visualize cost curve. Generates a plot that gives an explicit representation of the expected cost, as described by [1]. Options are greyed out if they do not apply to the specic set of results. 5 Clustering Selecting a Clusterer By now you will be familiar with the process of selecting and conguring objects. Clicking on the clustering scheme listed in the Clusterer box at the top of the window brings up a GenericObjectEditor dialog with which to choose a new clustering scheme. Cluster Modes The Cluster mode box is used to choose what to cluster and how to evaluate the results. The rst three options are the same as for classication: Use training set, Supplied test set and Percentage split (Section 4)except that now the data is assigned to clusters instead of trying to predict a specic class. The fourth mode, Classes to clusters evaluation, compares how well the chosen clusters match up with a pre-assigned class in the data. The dropdown box below this option selects the class, just as in the Classify panel. An additional option in the Cluster mode box, the Store clusters for visualization tick box, determines whether or not it will be possible to visualize the clusters once training is complete. When dealing with datasets that are so large that memory becomes a problem it may be helpful to disable this option. Ignoring Attributes Often, some attributes in the data should be ignored when clustering. The Ignore attributes button brings up a small window that allows you to select which attributes are ignored. Clicking on an attribute in the window highlights it, holding down the SHIFT key selects a range of consecutive attributes, and holding down CTRL toggles individual attributes on and o. To cancel the selection, back out with the Cancel button. To activate it, click the Select button. The next time clustering is invoked, the selected attributes are ignored. 9 Learning Clusters The Cluster section, like the Classify section, has Start/Stop buttons, a result text area and a result list. These all behave just like their classication counterparts. Right-clicking an entry in the result list brings up a similar menu, except that it shows only two visualization options: Visualize cluster assignments and Visualize tree. The latter is grayed out when it is not applicable. 6 Associating Setting Up This panel contains schemes for learning association rules, and the learners are chosen and congured in the same way as the clusterers, lters, and classiers in the other panels. Learning Associations Once appropriate parameters for the association rule learner bave been set, click the Start button. When complete, right-clicking on an entry in the result list allows the results to be viewed or saved. 7 Selecting Attributes Searching and Evaluating Attribute selection involves searching through all possible combinations of attributes in the data to nd which subset of attributes works best for prediction. To do this, two objects must be set up: an attribute evaluator and a search method. The evaluator determines what method is used to assign a worth to each subset of attributes. The search method determines what style of searc...

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Interest RatesChapter 4Options, Futures, and Other Derivatives 6th Edition, Copyright John C. Hull 20054.1Types of Rates Treasuryrates LIBOR rates Repo ratesOptions, Futures, and Other Derivatives 6th Edition, Copyright John C. Hull 2
U. Memphis - FIR - 7721
Wiener Processes and It's LemmaChapter 12Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005Types of Stochastic Processes Discretetime; discrete variable Discrete time; continuous variable Continuous time; dis
U. Memphis - FIR - 7721
Options on Stock Indices, Currencies, and FuturesChapter 14Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 200514.1European Options on Stocks Providing a Dividend YieldWe get the same probability distribution for
U. Memphis - FIR - 7721
Hedging Strategies Using FuturesChapter 3Options, Futures, and Other Derivatives 6th Edition, Copyright John C. Hull 20053.1Long & Short HedgesAlong futures hedge is appropriate when you know you will purchase an asset in the future and wa
U. Memphis - FIR - 7721
The Greek LettersChapter 15Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 200515.1ExampleA bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock S = 49, K = 50, r = 5
Auburn - ELEC - 4200
KCPSM38-bit Micro Controller for Spartan-3, Virtex-II and Virtex-IIPROFor Spartan-II(E) and Virtex(E) please use KCPSM Virtex-II and Virtex-IIPro are also supported by KCPSM2Ken Chapman Xilinx Ltd October 2003Rev.7ContentsUnderstanding KCPSM
WVU - RESM - 440
The Universal Transverse Mercator (UTM) GridMap ProjectionsThe most convenient way to identify points on the curved surface of the Earth is with a system of reference lines called parallels of latitude and meridians of longitude. On some maps, the
UVA - ASTR - 511
Getting Started with IDLIDL Version 6.0 July, 2003 EditionCopyright Research Systems, Inc. All Rights Reserved0703IDL60GSRestricted Rights NoticeThe IDL, ION ScriptTM, and ION JavaTM software programs and the accompanying procedures, functio
Ole Miss - CS - 490
EActiveState PerlE.1 IntroductionWhile Perl was initially developed on the UNIX platform, it was always intended to be a cross-platform computer language. ActivePerl is a version of Perl for Windows. The latest version of ActivePerl, the Perl 5.6
UGA - BCMB - 8020
REVIEWAssembly of Cell Regulatory Systems Through Protein Interaction DomainsTony Pawson1,2* and Piers Nash1 The sequencing of complete genomes provides a list that includes the proteins responsible for cellular regulation. However, this does not i
Montana - MB - 437
JOURNAL OF VIROLOGY, Aug. 2000, p. 70797084 0022-538X/00/$04.00 0 Copyright 2000, American Society for Microbiology. All Rights Reserved.Vol. 74, No. 15A Hypothesis for DNA Viruses as the Origin of Eukaryotic Replication ProteinsLUIS P. VILLARR
UCSB - ECE - 124
Errata for the Dally/Poulton "Digital Systems Engineering" Text.This list compiled by Fred Rosenberger (fred@cse.wustl.edu, http:/www.cse.wustl.edu/~fred ) as an aid to anyone using the Dally/Poulton text. I expect some of the "errors" reported here
N.C. State - CSC - 405
"" % &' () ' '$#! $# *'+ ,+-+./, # '34 ! ! ! ! " ! #$ '34 3 2 '34 6 $ 0 1 ! 5, % % & !07 .1,'
CSU Bakersfield - FIN - 600
18 - 1Distributions to Shareholders: Dividends and Repurchases Theories of investor preferences Signaling effects Residual model Dividend reinvestment plans Stock dividends and stock splits Stock repurchasesCopyright 2002 by Harcourt Inc. A
CSU Bakersfield - FIN - 600
13 - 1CHAPTER 13The Basics of Capital Budgeting: Evaluating Cash FlowsShould we build this plant?Copyright 2002 Harcourt, Inc.All rights reserved.13 - 2What is capital budgeting? Analysis of potential additions to fixed assets. Long-t
CSU Bakersfield - FIN - 600
20 - 1CHAPTER 20Lease Financing Types of leases Tax treatment of leases Effects on financial statements Lessee's analysis Lessor's analysis Other issues in lease analysisCopyright 2002 Harcourt, Inc. All rights reserved.20 - 2Who are t
CSU Bakersfield - FIN - 600
24 - 1CHAPTER 24Derivatives and Risk Management Risk management and stock value maximization. Derivative securities. Fundamentals of risk management. Using derivatives to reduce interest rate risk.Copyright 2002 Harcourt, Inc. All rights res
CSU Bakersfield - FIN - 600
12 - 1CHAPTER 12Corporate Valuation and ValueBased Management Corporate Valuation Value-Based Management Corporate GovernanceCopyright 2002 Harcourt, Inc.All rights reserved.12 - 2Corporate Valuation: List the two types of assets that
CSU Bakersfield - FIN - 600
10 - 1CHAPTER 10Stocks and Their Valuation Features of common stock Determining common stock values Efficient markets Preferred stockCopyright 2002 Harcourt, Inc. All rights reserved.10 - 2Facts about Common Stock Represents ownership.
Nevada - BADM - 720
JOHN S. HAMMONDLearning by the Case MethodSimply stated, the case method calls for discussion of real-life situations that business executives have faced. Casewriters, as good reporters, have written up these situations to present you with the in
UNC - MATH - 524
1 10.8 0.80.6 0.60.4 0.40.2 0.2-20 -20 -10 10 20-101020-0.2 -0.2Figure 1: Original function: f (x) =sin x x.Figure 2: Approximation by the Taylor polynomial of order 2110.80.80.60.60.40.40.20.2-20-101
UNC - MATH - 383
Math 302 - Dierential Equations (Metcalfe)Summer 2001 June 18, 2001Method of Undetermined Coecients (Section 3.6, 4.3) When: Use this technique to solve linear nonhomogeneous equations when the forcing term consists of combinations of polynomials,
UNC - MATH - 383
Math 302 - Dierential Equations (Metcalfe)Summer 2001 June 5, 2001Reduction of Order When: We know that the general solution of a second-order, linear homogeneous dierential equation consists of two independent pieces. If we know one of these two
UNC - MATH - 383
Math 302 - Dierential Equations (Metcalfe)Summer 2001 June 3, 2001Solving Linear, Homogeneous Second-Order Equations with Constant Coecients (Sections 3.1,3.4 When: Use this technique for linear homogeneous second-order equations with constant coe
UNC - MATH - 383
Math 302 - Dierential Equations (Metcalfe)Summer 2001 May 30, 2001Solving First Order Linear Equations using Integrating Factors (Section 2.1) When: Use this technique for rst-order linear equations. What: We will multiply the entire equation by a
UNC - MATH - 383
Math 302 - Dierential Equations (Metcalfe)Summer 2001 June 1, 2001Solving Exact Dierential Equations (Section 2.6) When: Use this technique for rst-order exact equations. If you write your rst-order (ordinary) dierential equation in the form M (x,