{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {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 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 1 }{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 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 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 1 }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 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 11 "Math 2250-3" }}{PARA 257 "" 0 " " {TEXT -1 22 "Friday August 22, 2003" }}{PARA 258 "" 0 "" {TEXT -1 55 "First order differential equations - notes for %1.1-1.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "HW for Wednesda y August 27: (bold means hand in, others optional)" }}{PARA 0 "" 0 "" {TEXT -1 19 " %1.1: 3, 4, " }{TEXT 264 1 "7" }{TEXT -1 6 ", 15, \+ " }{TEXT 265 2 "16" }{TEXT -1 6 ", 19, " }{TEXT 266 2 "20" }{TEXT -1 6 ", 27, " }{TEXT 267 2 "28" }{TEXT -1 6 ", 33, " }{TEXT 268 2 "35" } {TEXT -1 2 ", " }{TEXT 269 2 "36" }}{PARA 0 "" 0 "" {TEXT -1 16 " \+ %1.2: 5, " }{TEXT 270 1 "7" }{TEXT -1 2 ", " }{TEXT 271 2 "10" } {TEXT -1 6 ", 13, " }{TEXT 272 2 "14" }{TEXT -1 10 ", 19, 21, " } {TEXT 273 2 "25" }{TEXT -1 2 ", " }{TEXT 274 2 "26" }{TEXT -1 2 ", " } {TEXT 275 2 "29" }{TEXT -1 2 ", " }{TEXT 276 2 "30" }{TEXT -1 2 ", " } {TEXT 277 2 "36" }}{PARA 0 "" 0 "" {TEXT -1 16 " %1.3: 3, " } {TEXT 278 1 "6" }{TEXT -1 2 ", " }{TEXT 279 1 "8" }{TEXT -1 6 ", 11, \+ " }{TEXT 280 2 "12" }{TEXT -1 2 ", " }{TEXT 281 2 "16" }{TEXT -1 2 ", \+ " }{TEXT 282 2 "21" }{TEXT -1 2 ", " }{TEXT 283 2 "24" }{TEXT -1 8 ", \+ 32, 33" }}{PARA 0 "" 0 "" {TEXT -1 18 "Course home page: " }{URLLINK 17 "http://www.math.utah.edu/~korevaar/2250fall03" 4 "http://www.math. utah.edu/~korevaar/2250fall03" "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Tuesd ay problem sessions: 7:30-8:20 a.m. in EMCB 101; 10:45-11:35 a.m. in LS 101; " }}{PARA 0 "" 0 "" {TEXT -1 57 " 12:55-1:45 p.m. in JWB 335; 5-5:50 p.m. in LCB 152." }}{PARA 0 "" 0 "" {TEXT -1 92 "Maple a nd Math lab introductions: All are in LCB 115. Tuesday August 26: 11 :50-12:40, 2-3;" }}{PARA 0 "" 0 "" {TEXT -1 43 " Wednesday August \+ 27: 11:50-12:40, 5-6." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "What is a differential equation? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "What is the order of a di fferential equation?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 120 "How do you transform a sentence about rates of change \+ for a function into a differential equation? Can you do %1.1 #34?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "How do you che ck whether a function solves a differential equation? Can you do %1.1 #4? Or see example 2 below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 256 34 "First order differential equa tion " }{TEXT -1 40 " is any equation which can be written as" }} {EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6%%\"xG%\"yG*&%#dyG\"\" \"%#dxG!\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 13 "For example, " }} {EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"yG\"\"\"*$)7#*&%#dyGF&%#d xG!\"\"\"\"#F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 92 "is a differential \+ equation. Often we can rewrite a general D.E. in the more convenient \+ form" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#dyG\"\"\"%#dxG!\" \"-%\"fG6$%\"xG%\"yG" }}}{PARA 0 "" 0 "" {TEXT -1 220 "Can you do so f or the example above? No matter how we write the differential equatio n, the goal is to understand the functions y(x) which make it true. S ometimes we want the particular solution y(x) which satisfies an " } {TEXT 258 17 "initial condition" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#&%\"xG6#\"\"!&F%F)" }}}{PARA 0 "" 0 "" {TEXT -1 44 "The n the corresponding problem is called an " }{TEXT 257 21 "initial valu e problem" }{TEXT -1 265 ". The reason for the name \"initial conditi on\" is that our independent variable will often be time \"t\" rather \+ than x, and the initial condition will be the value of the solution fu nction at time t=0, for example an initial population in a population \+ growth problem." }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 120 "In section 1 .2 the book focuses on the easiest kind first order differential equat ions to solve, namely ones of the form" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 158 "Notice the unknown function y(x) does not appear on t he right hand side of this D.E so you are simply looking for antider ivatives of f(x) with respect to x:" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&-%$IntG6$-%\"fG6#%\"xGF,\"\"\"%\"CGF-" }}} {PARA 0 "" 0 "" {TEXT -1 254 " You got good at antidifferentiation i n Calculus!!!! Basic graviational attraction problems in physics (lik e the one we did yesterday) are of this form, when \"t\" is the variab le. Sometimes more complicated problems also reduce to antidifferenti ation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 13 "Exa mple 1. " }{TEXT 259 71 "Solve this initial value problem, and then \+ graph the solution function:" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6 #/*&%#dyG\"\"\"%#dxG!\"\",&%\"xGF&!\"$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "Here's how Maple would solve the prob lem:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "with(DEtools): #loa d differential equations package" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "dsolve(\{diff(y(x),x)=x-3,y(1)=2\},y(x));" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 53 "You can buy Maple from the bookstore for around $100." }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 12 "Example 2. " }{TEXT -1 32 "Solve the initial value problem:" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6# /*&%#dyG\"\"\"%#dxG!\"\",&%\"yGF&%\"xGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"!F'" }}}{PARA 0 "" 0 "" {TEXT -1 323 "Hint: You CA NNOT solve this problem by pure antidifferentiation since you can't an tidifferentiate the y(x) on the right side if you don't know it yet. \+ We WILL learn an algorithm for solving this sort of DE in section 1.4. For today you may use the fact that the general solution to this dif ferential equation is given by" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(F'\"\"\"F)F)*&%\"CGF))%\"eGF'F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 43 "Step 1: verify that y(x) does solve the DE." }}{PARA 0 "" 0 "" {TEXT -1 67 "Step 2: find a value of C in the general soluti on to solve the IVP." }}{PARA 0 "" 0 "" {TEXT -1 35 "Step3: graph the \+ solution function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT -1 12 "Slope Fields" }}{PARA 0 "" 0 "" {TEXT -1 403 "And now for something which seems completely different ( but isn't). The geometric interpretation of first order differential equations is connected with slope fields. (The use of the word \"fie ld\" is analogous to how it's used with the prefixes \"force\", \"elec tric\", \"magnetic\", \"wheat\".) Slope fields will be discussed in m ore detail on Monday, and we introduce them now. The differential eq uation " }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#dyG\"\"\"%#dxG! \"\"-%\"fG6$%\"xG%\"yG" }}}{PARA 0 "" 0 "" {TEXT -1 251 "is telling us that the slope of the solution graph at a point (x,y(x)), is given by the function f(x,y). We can use this fact to approximate solutions, \+ especially when exact formulas cannot be found: By hand or computer w e can draw a picture of the " }{TEXT 261 11 "slope field" }{TEXT -1 284 " determined by f, i.e. at each point (x,y) assign a slope using t he formula f(x,y). The natural way to indicate these slopes is with s mall line segments, see below. Solution graphs to the differential eq uation will be tangent to these slope fields. If there is an initial \+ condition" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#&%\"xG6# \"\"!&F%F)" }}}{PARA 0 "" 0 "" {TEXT -1 51 "then the solution graph wi ll pass through the point" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#7$ &%\"xG6#\"\"!&%\"yGF&" }}}{PARA 260 "" 0 "" {TEXT -1 13 "Example 1. \+ " }{TEXT 262 26 "The initial value problem:" }}{EXCHG {PARA 11 "" 1 " " {XPPMATH 20 "6#/*&%#dyG\"\"\"%#dxG!\"\",&%\"xGF&!\"$F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 23 "Picture of slope field:" }}{EXCHG {PARA 13 "" 1 "" {GLPLOT2D 420 294 294 {PLOTDATA 2 "6&-%'CURVESG6acn7$7$$!+X6)3=#!#6$!+cctX$*!#57 $$\"+X6)3=#F*$!+MkUl5!\"*7$7$$\"+,>$=:\"F-$!+15%*[$*F-7$$\"+*y)y1;F-$! +**e5l5F37$7$$\"+6H'4_#F-$!+Hue_$*F-7$$\"+p%yi*HF-$!+d7uk5F37$7$$\"+7- @*)QF-$!+$\\cnN*F-7$$\"+e=l'Q%F-$!+^VKk5F37$7$$\"+gaYc_F-$!+5'[:O*F-7$ $\"++t,ydF-$!+R^%Q1\"F37$7$$\"+D[gAmF-$!+&>)3n$*F-7$$\"+E')\\qrF-$!+!= 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color=black,linecolor=black,\n dirgrid=[30,30],\n title=`Slope field and solution for example 2`); \n #make a picture !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deqtnG/-%%diffG6$-%\"yG6#%\"xG F,,&F)\"\"\"F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG, (F'\"\"\"F)F)-%$expGF&!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 344 259 259 {PLOTDATA 2 "6(-%'CURVESG6acn7$7$$!33<%4\\)fw[?!#<$!3\"He!4:SB^HF*7$$! 3\"He!4:SB^>F*$!33<%4\\)fw[IF*7$7$$!3\")Ry(36%f2>F*$!35O[CCb>[HF*7$$!3 %Gkc,#Qa;=F*$!3*Q;bdZ/=0$F*7$7$$!3MMc,%\\\"om_F\"F*$!3\\Fo0;,NfIF*7$7$$!3lUvkG!=b?\"F*$!31e(Hd:*\\RHF*7$$!3E` $fvb4$R6F*$!3$>CqU%3]gIF*7$7$$!3HF2)>nbd1\"F*$!3s$o-'\\D`QHF*7$$!3F^1E X)4K+\"F*$!3F;tR]uYhIF*7$7$$!3+7:3^nih#*!#=$!3W`]B'*[rPHF*7$$!35+rnJxS p')F]q$!3bY\\w.^GiIF*7$7$$!356p4F6?nyF]q$!3XuH`M&=q$HF*7$$!3_Fl+mE@0tF ]q$!3aDqYl9)H1$F*7$7$$!38DX>\"35TZ'F]q$!3_?`]3;UOHF*7$$!3+SPDAIoRfF]q$ !3ZzY\\\"RyN1$F*7$7$$!3DW@x;K?#3&F]q$!3[(['o_n!f$HF*7$$!3][4-(>pHd%F]q 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