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186 Chapter 11: Differential Equation Graphing
11DIFFEQ.DOC TI-89/TI-92 Plus: Differential Equation (English) Susan Gullord Revised: 02/23/01 11:04 AM Printed: 02/23/01 2:15 PM Page 186 of 26
A system of equations can be defined in various ways, but the
following is a general method.
1. Rewrite the original differential
equation as necessary.
a. Solve for the highest-ordered
derivative.
b. Express it in terms of
y
and
t
.
c. On the right side of the equation
only, substitute to eliminate any
references to derivative values.
In place of: Substitute:
y
y'
y''
y'''
y
(4)
©
y1
y2
y3
y4
y5
©
d. On the left side of the equation,
substitute for the derivative value
as shown below.
In place of: Substitute:
y'
y''
y'''
y
(4)
©
y1'
y2'
y3'
y4'
©
2. On the applicable lines in the Y= Editor,
define the system of equations as:
y1' = y2
y2' = y3
y3' = y4
– up to –
y
n
' =
your n
th
-order equation
In a system such as this, the solution to the
y1'
equation is the
solution to the
n
th
-order equation. You may want to deselect any
other equations in the system.
Defining a System for Higher-Order Equations
In the Y= Editor, you must enter all differential equations as
1st-order equations. If you have an
n
th
-order equation, you
must transform it into a system of
n
1st-order equations.
Transforming an
Equation into a 1st-
Order System
Note:
To produce a 1st-
order equation, the right
side must contain non-
derivative variables only.
Note:
Based on the above
substitutions, the y' lines in
the Y= Editor represent:
y1' = y'
y2' = y''
etc.
So, this example’s 2nd-
order equation is entered on
the y2' line.
Do not
substitute on
the left side
at this time.
y'' + y' + y =
e
x
y'' =
e
x
ì
y'
ì
y
y'' =
e
t
ì
y'
ì
y
y'' =
e
t
ì
y2
ì
y1
y2' =
e
t
ì
y2
ì
y1
- TI-92 Plus 1
- TI-89 Shortcut Keys 3
- TI-92 Plus Shortcut Keys 4
- Table of Contents 5
- Flash Applications 12
- 00_front.doc 13
- Keystroke Differences 14
- What’s New? 16
- Upgradability with 17
- Flash ROM 17
- Custom Menu 17
- Chapter 1: 18
- Getting Started 18
- Installing the AAA 19
- Batteries 19
- Installing the AA 20
- Note: English cannot be 21
- Note: Until you complete 22
- Select a Language dialog 22
- About the Home 23
- Turning the 24
- TI-89 / TI-92 Plus 24
- Performing Computations 25
- 01START DOC TI 26
- Graphing a Function 28
- Chapter 2: 30
- Operating the Calculator 30
- Setting the Display Contrast 32
- The TI-89 Keyboard 33
- Modifier Keys 35
- Other Important 36
- Keys You Need to 36
- Be Familiar With 36
- Important Keys 37
- (continued) 37
- Home Screen 40
- Note: For an example of 41
- Entering Numbers 42
- Note: You can also select 44
- by using 44
- (page 44) 44
- Note: When you scroll to 45
- is displayed at 45
- Formats of Displayed Results 46
- Exact/Approx 47
- Display Digits Mode 48
- Exponential Format 48
- Displaying a Menu 51
- Selecting an Item 51
- Note: Because of limited 52
- Using the Custom Menu 54
- Selecting an Application 55
- F1 F2 F3 F4 F5 56
- WINDOW GRAPH TBLSet TABLEY= 56
- Setting Modes 57
- ¦ Performs 60
- Clean Up Toolbar 60
- Using the Catalog Dialog Box 61
- Indicated command 62
- Brackets [ ] indicate 62
- Note: Use the VAR-LINK 63
- Flash applications. See 63
- Chapter 21 63
- The variable’s value 65
- Tip: Reexecuting an entry 66
- Recalling a Previous 67
- Note: Refer to ans() in 68
- Appendix A 68
- ¦ You can use 69
- Why Use Auto-Paste 69
- Auto-Pasting an 69
- Entry or Answer 69
- Status Line 70
- Indicators 70
- Chapter 3: 74
- Symbolic Manipulation 74
- Automatic Simplification 81
- Functions that Use 83
- Simplification 83
- Tip: Use the solve function 85
- Specifying Domain 86
- Constraints 86
- Using Substitutions 86
- Note: For a complete 87
- Common Algebraic Operations 89
- Solving an Equation 90
- Solving a System of 90
- Linear Equations 90
- Overview of the Calc Menu 92
- Common Calculus Operations 93
- Define h(x)= 94
- (sin(t)/t,t,0,x) 94
- Multi-Statement vs 95
- Single-Statement 95
- Functions 95
- ˆ represents infinity 97
- Chapter 4: 98
- Entering Constants or Units 100
- Combining Multiple 101
- Using Parentheses 101
- Calculation 101
- For All Units Except 102
- Temperature 102
- For Temperature 103
- 04UNITS DOC TI 104
- Defaults for SI and 106
- Constants 106
- Chapter 5: 110
- Additional Home Screen Topics 110
- Saving the Entries 111
- Restoring the Saved 111
- Auto-paste vs 112
- Cut/Copy/Paste 112
- Cutting or Copying 112
- Information to the 112
- Clipboard 112
- Pasting Information 113
- Example: Copying 113
- Format of a 114
- Function 114
- Creating a User 114
- Defined Function 114
- Creating a Multi 115
- Statement Function 115
- Evaluating a 115
- Displaying and 116
- Editing a Function 116
- Definition 116
- Folders and 117
- Variables 117
- To cancel the menu or 118
- VAR-LINK 119
- Chapter 6: 122
- Basic Function Graphing 122
- ¦ Use the 124
- Graphing Functions 124
- Exploring the Graph 124
- Setting the Graph Mode 125
- POLAR graphing mode 126
- Selecting Functions to Graph 128
- Tip: To set 129
- as the style 129
- Defining the Viewing Window 130
- Changing the Graph Format 131
- indicator shows while 132
- ¦ Use a 133
- Free-Moving Cursor 133
- Tracing a Function 134
- Overview of the 136
- Zoom Menu 136
- Math Menu 139
- Tip: To erase a drawn 142
- (ReGraph) 142
- Chapter 7: 144
- Parametric Graphing 144
- PARAMETRIC 146
- Enter x and y components 147
- You can define 147
- Chapter 8: 150
- Polar Graphing 150
- Preview of Polar Graphing 151
- Window Variables 154
- Setting the Graph 154
- Chapter 9: 156
- Sequence Graphing 156
- Preview of Sequence Graphing 157
- Tip: You can also evaluate 158
- Simply enter the n value 158
- Selecting 160
- Sequences 160
- Selecting the 160
- Display Style 160
- Note: Both of these graphs 161
- ¦ To exit without making any 163
- Displaying the 163
- Dialog Box 163
- Using Web Plots 164
- Example: 165
- Convergence 165
- Divergence 165
- Oscillation 166
- Using Custom Plots 167
- Example: Fibonacci 168
- Sequence 168
- Chapter 10: 170
- 3D Graphing 170
- Preview of 3D Graphing 171
- You can define 3D 174
- Moving the Cursor in 3D 177
- The Viewing Orbit 181
- Animating the 181
- Animating a Series 181
- Tip: Setting Labels = ON is 182
- WIRE FRAME 183
- Contour Plots 184
- The default is 5. You can 185
- Tip: To remove the default 186
- Implicit Plots 188
- Chapter 11: 192
- Graphing 195
- Differential 195
- Equations 195
- Tip: You can use the 196
- Setting Axes 198
- Note: For information about 199
- Dimension 202
- Transforming an 203
- Equation into a 1st 203
- Order System 203
- Invalid Axes 204
- Undefined 206
- Predator-Prey Model 208
- Note: The fuzzy line on the 212
- Example of the deSolve( 213
- ) Function 213
- Fields=DIRFLD 215
- Fields=FLDOFF 216
- If You Use the Table 216
- Screen to View 216
- Chapter 12: 218
- Additional Graphing Topics 218
- for this 219
- Use the complete function 221
- Using the When 223
- Using a Multi 224
- Statement, User 224
- From the Home 224
- Screen or a Program 224
- Graphing a Family of Curves 225
- Using the Two-Graph Mode 226
- Note: You can display non 228
- (such as the Home screen) 228
- Drawing a Function 229
- Parametric, or Polar 229
- Equation 229
- Drawing the Inverse 229
- Tip: The text cursor 233
- Saving a Picture of 234
- Saving a Portion of 234
- Note: If a variable name is 235
- Note: Due to its complexity 236
- In two-graph mode 237
- Chapter 13: 238
- Preview of Tables 239
- Chapter 13: Tables 223 240
- Tip: To cancel a menu or 242
- Displaying an Automatic Table 243
- 228 Chapter 13: Tables 245
- TABLE SETUP 245
- Building a Manual (Ask) Table 246
- Note: If the independent 247
- Chapter 14: 248
- Split Screens 248
- Preview of Split Screens 249
- Note: In two-graph mode 250
- TOP-BOTTOM 253
- Chapter 15: 254
- Data/Matrix Editor 254
- Πmeans the cell is in a 255
- Data Variable 257
- Matrix Variable 257
- Creating a New 258
- Data, Matrix, or List 258
- Variable 258
- The Data/Matrix 260
- Editor Screen 260
- Entering or Editing 260
- Note: If you enter more than 261
- Changing the 262
- Cell Width 262
- Clearing a Column 262
- Using the Shift 267
- Using the CumSum 267
- Sorting Columns 268
- Chapter 16: 270
- Statistics and Data Plots 270
- Calculate 272
- FUNCTION 275
- The Calculate 276
- is shown instead of =, you 277
- Statistical Calculation Types 278
- Statistical Variables 280
- Defining a Statistical Plot 281
- Plot Type = Scatter 282
- Mark = Box 282
- Statistical Plot Types 283
- ¦ When you trace ( 284
- Histogram 284
- Showing the List of 285
- Stat Plots 285
- To Graph Plots and 285
- Y= Functions 285
- Type the CBL list 289
- Chapter 17: 292
- Programming 292
- Preview of Programming 293
- Running an Existing Program 295
- Starting a New 297
- Program or 297
- Note: By default, Variable 298
- Example of Passing 301
- Values to a Program 301
- Defines the subroutine 304
- Declares the subroutine 304
- Calls the subroutine 304
- Using Variables in a Program 305
- Circular Definition 306
- Variable-Related 306
- Commands 306
- :Local i 307
- :For i,0,5,1 307
- : Disp i 307
- ¦ Include 308
- To Perform 308
- Symbolic 308
- Calculations 308
- String Operations 309
- String Commands 310
- Conditional Tests 311
- Control Toolbar 314
- For...EndFor Loops 314
- While...EndWhile 315
- Note: The 316
- command 316
- Configuration 317
- Entering the 317
- SetMode Command 317
- I/O Toolbar Menu 318
- Input Commands 318
- Output Commands 319
- Graphical User 319
- Interface 319
- Creating a Custom Menu 320
- Restoring the 321
- Default Custom 321
- Note: For more information 322
- Chapter 12 323
- Drawing on the Graph Screen 324
- Drawing Lines and 325
- Expressions 325
- SendCalc 326
- Example 1 328
- Example 2 328
- Example 3 328
- Example 4 329
- Example 5 329
- Tip: If the program is not in 330
- Chapter 18: 332
- Text Editor 332
- Preview of Text Operations 333
- Note: By default 335
- Entering and Editing Text 336
- Thin cursor between 339
- Cursor highlights 339
- Entering Special Characters 341
- Note: To help you find the 342
- Note: Neither calculator 343
- For a List of All 344
- Special Characters 344
- Creating a Lab Report 347
- Printing the Report 348
- Chapter 19: 350
- Numeric Solver 350
- Preview of the Numeric Solver 351
- The argument does not have to 352
- Defining the Known Variables 354
- Finding the Solution 356
- Graphing the Solution 357
- Chapter 20: 360
- Number Bases 360
- Preview of Number Bases 361
- Domain error 362
- APPROXIMATE 363
- Boolean Operations 364
- Chapter 21: 366
- Notice that “ ( ” is 368
- indicates the 368
- Checking and Resetting Memory 370
- Flash ROM free on 371
- Which Applications 376
- Can You Use? 376
- Procedure 376
- From the VAR-LINK 378
- Archive free 380
- RAM free 380
- Chapter 22: 382
- Linking Two Units 383
- If you select: What happens: 385
- Note: The sending unit may 386
- Certificate 387
- Overview of 388
- The “Chat” Program 388
- Transferring 392
- Product Software 392
- (Base Code) 392
- TI-89 TI-89 TI-89 393
- Error Messages 394
- ID Lists and Group 395
- Certificates 395
- Collecting ID Lists 395
- Transmitting the ID 396
- List to a Computer 396
- Clearing the ID List 396
- Chapter 23: 400
- Activities 400
- Maximum Length of 401
- Pole in Hallway 401
- Hint: Use the auto-paste 402
- and press 402
- Performing 403
- Computations to 403
- Derive the Quadratic 403
- Note: This is only one of the 404
- Exploring a Matrix 405
- Exploring cos(x) = sin(x) 406
- Exploring a 3D 407
- Graph of the 407
- Surface Area of a 407
- Parallelepiped 407
- Finding the 408
- Minimum Surface 408
- Area Analytically 408
- Running a Tutorial 409
- 2:Clear split 410
- Decomposing a 411
- Rational Function 411
- Note: Be sure the Graph 412
- Filtering Data by 413
- Categories 413
- Note: Set up several box 414
- Note: Only Plot 1 through 415
- Plot 5 should be selected 415
- Hint: Press 417
- Optional Exercise 418
- Visualizing Complex 419
- Note: Calculating and 420
- Finding the Interest 421
- Rate of an Annuity 421
- Finding the Future 421
- Value of an Annuity 421
- TI-GRAPH 422
- Finding Factors 423
- Sampling-without 424
- Replacement 424
- Sampling without 424
- Appendix A: 426
- Functions and Instructions 426
- Quick-Find Locator 427
- Matrices 428
- Statistics 429
- ClrGraph 437
- CyclePic 446
- 8:Clear Home 448
- (creates the folder 451
- (deletes the folder 451
- DispHome 455
- DrawFunc 456
- DrawParm 456
- DropDown 457
- DrwCtour 458
- InputStr 475
- LineHorz 478
- LineVert 479
- {0 2 3 485
- PlotsOff 497
- PowerReg 498
- 8.124… 9.601… 11.078… 502
- 0. .904… 1.809… 502
- 0. 0. 0 502
- RandSeed 505
- SendChat 511
- Flash App 513
- Alternate Language 513
- ShowStat 517
- Unarchiv 533
- ZoomData 539
- ZoomPrev 541
- ZoomTrig 542
- , ', " 553
- Appendix B: 558
- Reference Information 558
- Note: For detailed 567
- Vector Format 569
- Pretty Print 569
- Split Screen 569
- Unit System 571
- Custom Units 571
- Language 571
- Disp char(127) 572
- TI-89 Key Codes 573
- ' 39 574
- " 34 8241 q 113 574
- Table 2: Arrow Keys 574
- Table 3: Greek Letters 575
- Key Modifier 576
- Table 6: Greek Letters 579
- Table 7: Tilde Letters 579
- Table 8: Caret Letters 579
- Table 9: Umlaut Letters 579
- Entering Complex Numbers 580
- Accuracy Information 583
- Order of Evaluation 585
- Parentheses 585
- Brackets, and 585
- Indirection 586
- Post Operators 586
- Exponentiation 586
- Negation 586
- Constraint ( 586
- Regression Formulas 587
- ¦ At each of the five points 589
- Algorithm 589
- Runge-Kutta Method 590
- Appendix C: 592
- Battery Information 593
- Lithium battery 594
- AAA alkaline 594
- Slide to open 595
- AA alkaline batteries 595
- In Case of Difficulty 596
- Home Page: 597
- education.ti.com 597
- Warranty Information 598
- Australia & New 599
- Zealand Customers 599
- All Other Customers 599
- Appendix D: 600
- Programmer’s Guide 600
- A (continued) 609
- C (continued) 610
- D (continued) 611
- E (continued) 612
- F (continued) 613
- G (continued) 614
- M (continued) 616
- P (continued) 618
- S (continued) 620
- U (continued) 622
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