{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "New Page" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 1 2 0 1 }{PSTYLE "Heading 1" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 1 {PARA 257 "" 0 "" {TEXT -1 91 "Boyce and DiPrima Eleme ntary Differential Equations\nChapter 2, Section 1 Selected Exercises " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Abstract" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Keywords" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 20 "Dr. Geor ge R. Barnes" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "How to Contact Dr . Barnes" }}{PARA 258 "" 0 "" {TEXT -1 29 "Home Page:................. " }{URLLINK 17 "http://www.math.louisville.edu/~grbarn01" 4 "http:// www.math.louisville.edu/~grbarn01" "" }{TEXT -1 2 " ." }}{PARA 258 "" 0 "" {TEXT -1 32 "405 Page:..................... " }{URLLINK 17 "http ://www.math.louisville.edu/~grbarn01/405/405home.html" 4 "http://www.m ath.louisville.edu/~grbarn01/405/405home.html" "" }{TEXT -1 2 " ." }} {PARA 258 "" 0 "" {TEXT -1 23 "405 Announcements:... " }{URLLINK 17 " http://www.math.louisville.edu/~grbarn01/405/405a.html" 4 "http://www. math.louisville.edu/~grbarn01/405/405a.html" "" }{TEXT -1 2 " ." }} {PARA 258 "" 0 "" {TEXT -1 22 "405 Maple Resources: " }{URLLINK 17 "h ttp://www.math.louisville.edu/~grbarn01/405/maple/_405_MapleResources. mws" 4 "http://www.math.louisville.edu/~grbarn01/405/maple/_405_MapleR esources.mws" "" }{TEXT -1 2 " ." }}{PARA 258 "" 0 "" {TEXT -1 4 " \+ " }}{PARA 258 "" 0 "" {TEXT -1 48 "email: george.barnes@l ouisville.edu" }}{PARA 258 "" 0 "" {TEXT -1 31 "Phone: 502 -852-2727" }}{PARA 258 "" 0 "" {TEXT -1 32 "FAX: 502-852 -7132 " }}{PARA 258 "" 0 "" {TEXT -1 32 "Dept: 502-852-6 826" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 40 " Mail: Dr. George R. Barnes" }}{PARA 258 "" 0 "" {TEXT -1 45 " Mathematics Depatment" }}{PARA 258 "" 0 "" {TEXT -1 58 " University of Louisville, KY 40292" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "About Math 405" }}{PARA 0 "" 0 "" {TEXT -1 262 "Mathemati cs 405, Differential Equations, is taught at least once per year.. Th is worksheet was prepared for the Spring 2001 term. The text book in \+ use was Elementary Differetial Equations 7th Ed., by Boyce and DiPrima . Contact Dr. Barnes for more information." }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 59 "(c) Copyright 2000 Dr. George R. Barnes and Dr. Inessa \+ Levi" }}{PARA 0 "" 0 "" {TEXT -1 308 "All Rights Reserved. This work \+ may not be translated or copied in whole or in part without the writte n permission of the copyright holders, except for brief excerpts in co nnection with scholarly analysis. Any use or modification of this wor ksheet, in whole or part, must prominently display this copyright." }} {PARA 15 "" 0 "" {TEXT -1 199 "Permission is hereby granted to Dr. Geo rge R. Barnes and to Dr. Inessa Levi to make editorial modifications \+ and to publish this worksheet and/or subsequent courses taught by Dr. Barnes or Dr. Levi." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Revision History" }}{PARA 15 "" 0 "" {TEXT -1 41 "2/18/2004 405_0201X_Exercises.mws created" }}{PARA 15 "" 0 "" {TEXT -1 36 "2/2/2004 Editorial subsections added" }}{PARA 15 "" 0 "" {TEXT -1 45 "4/17/01 editorial Name of file points to URL." }} {PARA 15 "" 0 "" {TEXT -1 55 "Revised 12/28/00 for Math 405 as _405BD7 _ccss_about.mws" }}{PARA 15 "" 0 "" {TEXT -1 52 "Revised 12/02/00 for \+ Math 405 as _405_ccss_about.mws" }}{PARA 15 "" 0 "" {TEXT -1 56 "Revis ed 10/17/00 for Math 206 as _206S4_ccss_about.mws ." }}{PARA 15 "" 0 " " {TEXT -1 43 "Created 6/22/00 as _301S4_ccss_about.mws . " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "About T his Worksheet " }{TEXT 259 46 "405_0201X_Exercises.mws last updated 2/ 18/2004" }}{PARA 15 "" 0 "" {TEXT -1 22 "Course Announcements: " } {URLLINK 17 "local" 4 "file:///c:/webshare/wwwroot/405/405a.html" "" } {TEXT -1 3 " , " }{URLLINK 17 "URL" 4 "http://www.math.louisville.edu/ ~grbarn01/405/405a.html" "" }{TEXT -1 35 " will open in a browserter w indow. " }}{PARA 15 "" 0 "" {TEXT -1 59 "Links to other Maple Resource s for Differential Equatons: " }{HYPERLNK 17 "_405_MapleResources.mws " 1 "_405_MapleResources.mws" "" }{TEXT -1 3 " , " }{URLLINK 17 "URL" 4 "www.math.louisville.edu/~grbarn01/405/maple/_405_MapleResources.mws " "" }{TEXT -1 3 ", " }{URLLINK 17 "Maple Directory Listing" 4 "www.m ath.louisville.edu/~grbarn01/405/maple/" "" }{TEXT -1 3 " . " }}{PARA 15 "" 0 "" {TEXT -1 31 "Save this file locally in your " }{HYPERLNK 17 "_405_MapleResources.mws (local)" 1 "_405_MapleResources.mws" "" } {TEXT -1 14 " directory as " }{TEXT 256 23 "405_0201X_Exercises.mws" } {TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 89 "Insert the following b ullet in other files to refer to this file. Modify the update time." } }{PARA 15 "" 0 "" {URLLINK 17 "405_0201X_Exercises.mws" 4 "www.math.lo uisville.edu/~grbarn01/405/maple/405_0201X_Exercises.mws" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "local" 1 "405_0201X_Exercises.mws" "" }{TEXT -1 2 ", " }{URLLINK 17 "URL" 4 "www.math.louisville.edu/~grbarn01/405/ maple/405_0201X_Exercises.mws" "" }{TEXT -1 135 " Updated 2/18/04. Exe rcise 21 (direction fields, initial condition analysis). The URL point s to the most recent version of this file. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Discussion" }}{PARA 0 "" 0 "" {TEXT -1 18 "See the wor ksheet " }}{PARA 15 "" 0 "" {HYPERLNK 17 "405BD7_0101_Models.mws" 1 "4 05BD7_0101_Models.mws" "" }{TEXT -1 3 " , " }{URLLINK 17 "URL" 4 "www. math.louisville.edu/~grbarn01/405/maple/405BD7_0101_Models.mws" "" } {TEXT -1 43 " Updated 01/05/2001 8:30p. Models and the " }{TEXT 257 10 "dfieldplot" }{TEXT -1 67 " command. The URL points to the most rec ent version of this file. " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" } }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Related Maple Commands " }}{PARA 0 "" 0 "" {TEXT -1 17 "See the worksheet" }}{PARA 15 "" 0 " " {HYPERLNK 17 "405BD7_0101_Models.mws" 1 "405BD7_0101_Models.mws" "" }{TEXT -1 3 " , " }{URLLINK 17 "URL" 4 "www.math.louisville.edu/~grbar n01/405/maple/405BD7_0101_Models.mws" "" }{TEXT -1 43 " Updated 01/05/ 2001 8:30p. Models and the " }{TEXT 258 10 "dfieldplot" }{TEXT -1 67 " command. The URL points to the most recent version of this file. " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "Exercise 21" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Exercise 21" }{TEXT -1 20 " y' -.5*y=2*cos(t)" }} {PARA 0 "" 0 "" {TEXT -1 40 "Examine the direction field with a plot \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "with(DEtools):(a,b):=(4* Pi,8);dfieldplot(D(y)(t)-.5*y(t)=2*cos(t), y(t), t=a..-a, y=-b..b);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"aG%\"bG6$,$*&\"\"%\"\"\"%#PiGF+ F+\"\")" }}{PARA 13 "" 1 "" {GLPLOT2D 693 363 363 {PLOTDATA 2 "6%-%'CU RVESG6^dl7%7$$!3A1p#\\\"zpw7!#;$!3;y=YTHy)f(!#<7$$!3%R4t!4idO7F*$!3&47 Q&eq@,%)F-7$$!37&=\\h02&H7F*$!3/X+$p8db>)F-7%7$$!3omK/qe;O6F*$!3#*Q\\v D-r&e(F-7$$!3%[O4I3`D6\"F*$!33h]Cu(*G9%)F-7$$!3#)=\"o*ziR,6F*$!3b;#e3e 1h@)F-7%7$$!3a$3#Q;!*[$***F-$!3S\"))*ovm\\\"e(F-7$$!3UX0n/)[\"[)*F-$!3 \\>,JCL]=%)F-7$$!3SFdL7+F:(*F-$!3_`Hai-sD#)F-7%7$$!3[y(=tYFhn)F-$!3/X# >v3#*=e(F-7$$!3Dk,EW'e*>&)F-$!3'\\v![7z5=%)F-7$$!3Uoi\"Hfz&*Q)F-$!3y9s 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"" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 46 "Observations about the fieldplot and solutions" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "`y '` = .5*y+2*cos(t);" "6#/% $y~'G,&*&-%&FloatG6$\"\"&!\"\"\"\"\"%\"yGF,F,*&\"\"#F,-%$cosG6#%\"tGF, F," }{TEXT -1 18 " is positive when " }{XPPEDIT 18 0 "4 < y;" "6#2\"\" %%\"yG" }{TEXT -1 30 " so once y > 4 y(t) increases." }}{PARA 0 "" 0 " " {TEXT -1 75 "Similarly y ' is negative if y < -4 and hence y will de crease even further." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "dsolve(\{D(y)(t)-.5*y(t)=2*cos(t),y(0)=z \},y(t));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,(*&#\"\"%\"\"&\"\"\"-%$cosGF&F-!\"\"*&# \"\")F,F--%$sinGF&F-F-*&-%$expG6#,$*&\"\"#F0F'F-F-F-,&#F+F,F-%\"zGF-F- F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 50 "Observation on th e effect of the initial condition" }}{PARA 0 "" 0 "" {TEXT -1 21 "When y(0) = z =-.8, " }{XPPEDIT 18 0 "z+4/5 = 0;" "6#/,&%\"zG\"\"\"*&\"\" %F&\"\"&!\"\"F&\"\"!" }{TEXT -1 138 " so there will be no exponential, otherwise the sign of the exponential term depends on whether the ini tial value z is above or below -.8." }}{PARA 0 "" 0 "" {TEXT -1 72 "Th e only bounded solution corresponds to the initial condition z = -0.8. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Further activities" }} {PARA 15 "" 0 "" {TEXT -1 22 "Course Announcements: " }{URLLINK 17 "lo cal" 4 "file:///c:/webshare/wwwroot/405/405a.html" "" }{TEXT -1 3 " , \+ " }{URLLINK 17 "URL" 4 "http://www.math.louisville.edu/~grbarn01/405/4 05a.html" "" }{TEXT -1 32 " will open in a browser window. " }}{PARA 15 "" 0 "" {TEXT -1 51 "Links to other Maple Resources for Calculus II I: " }{HYPERLNK 17 "_405_MapleResources.mws" 1 "_405_MapleResources. mws" "" }{TEXT -1 3 " , " }{URLLINK 17 "URL" 4 "www.math.louisville.ed u/~grbarn01/405/maple/_405_MapleResources.mws" "" }{TEXT -1 3 ", " } {URLLINK 17 "Maple Directory Listing" 4 "www.math.louisville.edu/~grba rn01/405/maple/" "" }}{PARA 15 "" 0 "" {TEXT -1 23 "Template for new t opic " }{HYPERLNK 17 "_405BD7_ccss_about.mws" 1 "_405BD7_ccss_about.mw s" "" }{TEXT -1 4 " , " }{URLLINK 17 "Web" 4 "www.math.louisville.edu /~grbarn01/405/maple/_405BD7_ccss_about.mws" "" }{TEXT -1 3 " . " }}}} {MARK "5 3 12 2 0" 72 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }