{VERSION 6 1 "Windows XP" "6.1" } {USTYLETAB {PSTYLE "_pstyle33" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "H eading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle3" -1 202 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Left Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle40" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle21" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "C ourier" 1 9 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle10" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnostic " -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle61" -1 206 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle1" -1 207 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE " _pstyle47" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle54" -1 209 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 210 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "_pstyle15" -1 211 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 12 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle8" -1 212 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle18" -1 213 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle59" -1 214 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE " _pstyle48" -1 215 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle7" -1 216 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 12 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_ pstyle56" -1 217 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Annotation Title" -1 218 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle28" -1 219 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 12 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle63" -1 220 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle 58" -1 221 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle23" -1 222 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_ pstyle12" -1 223 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 12 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle2" -1 224 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 0 0 1 }3 1 0 0 4 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle35" -1 225 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle5" -1 226 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle31" -1 227 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle50" -1 228 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle65" -1 229 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle 36" -1 230 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle43" -1 231 1 {CSTYLE " " -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fix ed Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle14" -1 232 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 12 0 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle9" -1 233 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 12 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle20" -1 234 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle29" -1 235 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle 64" -1 236 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle55" -1 237 1 {CSTYLE " " -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle24" -1 238 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle19" -1 239 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle17" -1 240 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle38" -1 241 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle49" -1 242 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered Lis t 5" -1 243 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle62" -1 244 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 12 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle27" -1 245 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle16" -1 246 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 12 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle39 " -1 247 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle11" -1 248 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 12 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle46" -1 249 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle32" -1 250 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle26" -1 251 1 {CSTYLE "" -1 -1 "Time s" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 } {PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle37" -1 252 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 } 1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle60" -1 253 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle25" -1 254 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_psty le30" -1 255 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle66" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle13" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 12 0 2 0 2 0 2 2 -1 1 }{PSTYLE " _pstyle44" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle52" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle51" -1 260 1 {CSTYLE "" -1 -1 "Time s" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle34" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle45" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 12 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 4" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 }{PSTYLE " _pstyle57" -1 264 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle22" -1 265 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle6" -1 266 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line \+ Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Cou rier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle41" -1 267 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle 42" -1 268 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle4" -1 269 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 12 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 2" -1 270 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 } {PSTYLE "_pstyle53" -1 271 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle272" -1 200 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle275" -1 201 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle2" -1 202 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle270" -1 203 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Nontermin al" -1 24 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle 4" -1 204 "Times" 1 16 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "Maple Com ment" -1 21 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_csty le5" -1 205 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle 32" -1 206 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle1 0" -1 207 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle29 " -1 208 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle271 " -1 209 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle35" -1 210 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "Annotation Tex t" -1 211 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Unde rlined Bold" -1 41 "Times" 1 12 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE " _cstyle34" -1 212 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_ cstyle47" -1 213 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Pa ragraphStyle1" -1 214 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle17" -1 215 "Times" 1 16 0 0 0 1 2 1 1 2 2 2 0 0 0 1 } {CSTYLE "_cstyle36" -1 216 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle258" -1 217 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle31" -1 218 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle274" -1 219 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle30" -1 220 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle65" -1 221 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "Dictionary Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle257" -1 222 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle276" -1 223 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle3" -1 224 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle56" -1 225 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle256" -1 226 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle55" -1 227 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle14" -1 228 "Times" 1 16 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle62" -1 229 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle40" -1 230 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle58" -1 231 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle9" -1 232 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle42" -1 233 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle45" -1 234 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle7" -1 235 "Times" 1 16 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle262" -1 236 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "ParagraphStyle2" -1 237 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle266" -1 238 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle46" -1 239 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle8" -1 240 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle13" -1 241 "Times" 1 16 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "ParagraphStyle3" -1 242 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle63" -1 243 "Time s" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle59" -1 244 " Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle61" -1 245 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle11" -1 246 "Times" 1 16 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle33" -1 247 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle259" -1 248 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Output Label s" -1 29 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Emphas ized280" -1 249 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cs tyle43" -1 250 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cst yle27" -1 251 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2 D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_ cstyle28" -1 252 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Pr ompt" -1 1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyl e26" -1 253 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle 41" -1 254 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cst yle60" -1 255 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_csty le12" -1 256 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help V ariable" -1 25 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_c style19" -1 257 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hel p Bold" -1 39 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Page \+ Number" -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Defau lt" -1 38 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math S mall" -1 7 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Inert Output" -1 258 "Times" 1 12 144 144 144 1 2 2 2 2 1 2 0 0 0 1 } {CSTYLE "_cstyle265" -1 259 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "Popup" -1 31 "Times" 1 12 0 128 128 1 1 2 1 2 2 2 0 0 0 1 } {CSTYLE "_cstyle18" -1 260 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "Maple Input Placeholder" -1 261 "Courier" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle24" -1 262 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle268" -1 263 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle261" -1 264 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle269" -1 265 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle260" -1 266 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small206" -1 267 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle20" -1 268 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle64" -1 269 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle267" -1 270 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle6" -1 271 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle264" -1 272 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 22 "Ti mes" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle21" -1 273 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined" -1 44 "Times " 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times " 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle263" -1 274 "Times " 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle44" -1 275 "Courie r" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle25" -1 276 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle39" -1 277 "Cour ier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "Text" -1 278 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle273" -1 279 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle53" -1 280 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small" -1 281 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle23" -1 282 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle48" -1 283 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle16" -1 284 "Times" 1 16 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle54" -1 285 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle66" -1 286 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle22" -1 287 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle4 9" -1 288 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Titl e" -1 27 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle277 " -1 289 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle57" -1 290 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle278" -1 291 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle50" -1 292 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Equation Lab el" -1 293 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle1 5" -1 294 "Times" 1 16 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle52 " -1 295 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle1" -1 296 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle38" -1 297 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle51" -1 298 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Maple Na me" -1 35 "Times" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help \+ Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyl e37" -1 299 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {PARA 242 "" 0 "" {TEXT 280 57 "Euler's Method of Solving Ord inary Differential Equations" }{TEXT 280 0 "" }}{PARA 228 "" 0 "" {TEXT 285 25 "Nathan Collier, Autar Kaw" }{TEXT 285 0 "" }}{PARA 228 " " 0 "" {TEXT 285 27 "University of South Florida" }{TEXT 285 0 "" }} {PARA 228 "" 0 "" {TEXT 285 24 "United States of America" }{TEXT 285 0 "" }}{PARA 228 "" 0 "" {TEXT 285 15 "kaw@eng.usf.edu" }{TEXT 285 0 " " }}{SECT 0 {PARA 260 "" 0 "" {TEXT 227 12 "Introduction" }{TEXT 227 0 "" }}{PARA 259 "" 0 "" {TEXT 225 118 "This worksheet demonstrates th e use of Maple to illustrate Euler's method of solving ordinary differ ential equations. " }{TEXT 225 0 "" }}{PARA 259 "" 0 "" {TEXT 225 0 "" }}{PARA 259 "" 0 "" {TEXT 225 198 "Euler's method of solving ordinary differential equations uses the derivative and value of the function \+ at the initial condition to project the location and value of the next point on the function. " }{TEXT 225 0 "" }}{PARA 271 "" 0 "" {XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mr ow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mfrac(Typeset ting:-mrow(Typesetting:-mo(\"ⅆ\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\")), Typesetti ng:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mo(\"&D ifferentialD;\", mathvariant = \"normal\", fence = \"false\", separato r = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), linethickness = \+ \"1\", denomalign = \"center\", numalign = \"center\", bevelled = \"fa lse\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\" , fence = \"false\", separator = \"false\", stretchy = \"false\", symm etric = \"false\", largeop = \"false\", movablelimits = \"false\", acc ent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting: -mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic \"), Typesetting:-mo(\"⁡\", mathvariant = \"normal\", fe nce = \"false\", separator = \"false\", stretchy = \"false\", symmetri c = \"false\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfe nced(Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathv ariant = \"italic\")), mathvariant = \"normal\")), Typesetting:-mi(\" \")), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow (Typesetting:-mi(\"f\", italic = \"true\", mathvariant = \"italic\"), \+ Typesetting:-mo(\"⁡\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced (Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvaria nt = \"italic\"), Typesetting:-mo(\",\", mathvariant = \"normal\", fen ce = \"false\", separator = \"true\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.0em\", rspace = \"0.3333333em\"), Typesetting: -mi(\"y\", italic = \"true\", mathvariant = \"italic\")), mathvariant \+ = \"normal\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%m rowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6' F+-F#6&-I&mfracGF$6(-F#6#-I#moGF$6-Q0ⅆF'/%,mathvariantGQ 'normalF'/%&fenceGQ&falseF'/%*separatorGFA/%)stretchyGFA/%*symmetricGF A/%(largeopGFA/%.movablelimitsGFA/%'accentGFA/%'lspaceGQ&0.0emF'/%'rsp aceGFP-F#6%F+-F#6$F8-F,6%Q\"xF'/%'italicGQ%trueF'/F=Q'italicF'F+/%.lin ethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF^o/%)bevelledGFA -F96-Q1⁢F'F " 0 "" {MPLTEXT 1 231 8 "restart;" }{MPLTEXT 1 231 0 "" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 227 16 "Section 1: Inpu t" }{TEXT 227 0 "" }}{PARA 259 "" 0 "" {TEXT 225 74 "The following is \+ the data used in solving ordinary differential equations:" }{TEXT 225 0 "" }}{EXCHG {PARA 217 "" 0 "" {TEXT 290 34 "Differential equation of the form " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Ty pesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:- mfrac(Typesetting:-mrow(Typesetting:-mo(\"ⅆ\", mathvaria nt = \"normal\", fence = \"false\", separator = \"false\", stretchy = \+ \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\" )), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typeset ting:-mo(\"ⅆ\", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", ital ic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), lin ethickness = \"1\", denomalign = \"center\", numalign = \"center\", be velled = \"false\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \" false\", symmetric = \"false\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\") , Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvari ant = \"italic\"), Typesetting:-mo(\"⁡\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Ty pesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \+ \"true\", mathvariant = \"italic\")), mathvariant = \"normal\")), Type setting:-mi(\"\")), Typesetting:-mo(\"=\", mathvariant = \"normal\", f ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typ esetting:-mrow(Typesetting:-mi(\"f\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"⁡\", mathvariant = \"nor mal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\" , accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typeset ting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true \", mathvariant = \"italic\"), Typesetting:-mo(\",\", mathvariant = \" normal\", fence = \"false\", separator = \"true\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.3333333em\") , Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\")) , mathvariant = \"normal\")), Typesetting:-mi(\"\")), Typesetting:-mi( \"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#mi GF$6#Q!F'-F#6'F+-F#6&-I&mfracGF$6(-F#6#-I#moGF$6-Q0ⅆF'/% ,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFA/%)stretchyGFA /%*symmetricGFA/%(largeopGFA/%.movablelimitsGFA/%'accentGFA/%'lspaceGQ &0.0emF'/%'rspaceGFP-F#6%F+-F#6$F8-F,6%Q\"xF'/%'italicGQ%trueF'/F=Q'it alicF'F+/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF^o /%)bevelledGFA-F96-Q1⁢F'F " 0 "" {MPLTEXT 1 244 28 "f:=(x, y)->y(x)*x^2-1.2*y(x);" }{MPLTEXT 1 244 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"f*6$I\"xG6\"I\"yG6\"6\"6$I)operatorG6\"I&arrow G6\"6\",&*&-I\"yG6\"6#I\"xG6\"\"\"\")I\"xG6\"\"\"#\"\"\"\"\"\"*&$\"#7! \"\"\"\"\"-I\"yG6\"6#I\"xG6\"\"\"\"!\"\"6\"6\"6\"" }{TEXT 20 0 "" }}} {EXCHG {PARA 217 "" 0 "" {TEXT 290 22 "Initial condition for " }{TEXT 255 1 "x" }{TEXT 290 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 8 "x0:=0.0:" }{MPLTEXT 1 244 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 290 22 "Initial condition for " }{TEXT 255 1 "y" }{TEXT 290 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 8 "y0:=1.0:" }{MPLTEXT 1 244 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 290 9 "Value of " }{TEXT 255 1 "x" }{TEXT 290 10 " at which " }{TEXT 255 1 "y" }{TEXT 290 11 " \+ is desired" }{TEXT 290 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 8 "xf:=2.0:" }{MPLTEXT 1 244 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 290 41 "This worksheet will find the solution at " }{TEXT 255 2 "xf" }{TEXT 290 80 " by taking 5 steps. What follows is the exact solu tion and each individual step." }{TEXT 290 0 "" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 227 25 "Section 2: Exact Solution" }{TEXT 227 0 "" }} {PARA 259 "" 0 "" {TEXT 225 81 "The following uses Maple's built-in to ols to find the exact solution for the ODE." }{TEXT 225 0 "" }}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 245 25 "ODE:=diff(y(x),x)=f(x,y);" } {MPLTEXT 1 245 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I$ODEG6\"/-I%d iffG%*protectedG6$-I\"yG6\"6#I\"xG6\"I\"xG6\",&*&-I\"yG6\"6#I\"xG6\"\" \"\")I\"xG6\"\"\"#\"\"\"\"\"\"*&$\"#7!\"\"\"\"\"-I\"yG6\"6#I\"xG6\"\" \"\"!\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 31 "soln:=(dsolve(\{ODE,y(x0)=y0\})):" }{MPLTEXT 1 244 0 "" }}} {EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 19 "assign(soln); y(x);" } {MPLTEXT 1 244 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#-I$expG6$%*prot ectedGI(_syslibG6\"6#,$*(#\"\"\"\"#:\"\"\"I\"xG6\"\"\"\",&*&\"\"&\"\" \")I\"xG6\"\"\"#\"\"\"\"\"\"\"#=!\"\"\"\"\"\"\"\"" }{TEXT 20 0 "" }}} {EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 102 "plot(y(x),x=x0..xf,titl e=\"Exact Solution Plotted on Given Interval\",legend=\"Exact y(x)\" , thickness=2);" }{MPLTEXT 1 244 0 "" }}{PARA 253 "" 1 "" {TEXT 300 0 "" }{GLPLOT2D 662 372 372 {PLOTDATA 2 "6)-%'CURVESG6%7W7$$\"\"!!\"\"$\"# 5!\"\"7$$\"1LLLL3VfV!#<$\"1k$>L@y0\\*!#;7$$\"1nmm\"H[D:)!#<$\"1:g=0Kmp !*!#;7$$\"2LLL$e0$=C\"!#<$\"1i\"\\uEB5i)!#;7$$\"2LLL$3RBr;!#<$\"1^%R.P jb>)!#;7$$\"2lmm\"zjf)4#!#<$\"0[F(oIu(z(!#:7$$\"2KLLe4;[\\#!#<$\"1#eIj 7g7X(!#;7$$\"2(****\\i'y]!H!#<$\"1x(Gtc$f9r!#;7$$\"1LL$ezs$HL!#;$\"1a4 vvmR*y'!#;7$$\"2&****\\7iI_P!#<$\"0)Hhs.x(['!#:7$$\"1nmm;_M(=%!#;$\"19 7lkC<+i!#;7$$\"2MLL$3y_qX!#<$\"1>/U'**G_'f!#;7$$\"*l+>+&!\"*$\"1c\\`=e e?d!#;7$$\"*vW]V&!\"*$\"1u(ya*pJ&\\&!#;7$$\"*NfC&e!\"*$\"1X?MZu#oH&!#; 7$$\"1LL$ez6:B'!#;$\"1p\"yvrP=8&!#;7$$\"1mmm;=C#o'!#;$\"2W1ep.,R&\\!#< 7$$\"1mmmm#pS1(!#;$\"29^N!e**==[!#<7$$\",DOD#3v!#6$\"1TSN&HPrn%!#;7$$ \"1mmmm(y8!z!#;$\"1K9om%zpc%!#;7$$\",DOIFL)!#6$\"0ErlzT;Y%!#:7$$\",v3z Mu)!#6$\"2$4U5t\"zhP%!#<7$$\"1nmm\"H_?<*!#;$\"2DZG*H;F-V!#<7$$\"1nm;zi hl&*!#;$\"2u47S'Q2[U!#<7$$\"1LLL3#G,***!#;$\"1bv+GA%!#<7$$\"2)***\\i&p@[7!#;$\"0'4)fEceF%!#:7$$ \"2)****\\2'HKH\"!#;$\"17F#ypikN%!#;7$$\"2lmmmZvOL\"!#;$\"1X?r,-,]W!#; 7$$\"+v+'oP\"!\"*$\"1PkLBW5uX!#;7$$\"2KL$eR<*fT\"!#;$\"28@`$GDO5Z!#<7$ $\"+&)Hxe9!\"*$\"1O-*Q!Q>))[!#;7$$\"2lm;H!o-*\\\"!#;$\"18\"HxBrj3&!#;7 $$\"2****\\7k.6a\"!#;$\"1\"\\+())RtH`!#;7$$\"2mmm;WTAe\"!#;$\"1Ta:;EU3 c!#;7$$\"-D1*3`i\"!#6$\"1wN`^m')\\f!#;7$$\"2MLLL*zym;!#;$\"1PZ/Q]MMj!# ;7$$\"2LLL3N1#4!#6$\"2$p4*Q6-n/\"!#;7$$ \"*PDj$>!\")$\"1(*[4x+?,6!#:7$$\"-v.Uac>!#6$\"2lAmA8r.;\"!#;7$$\".v=5s #y>!#7$\"16&>:*))pH7!#:7$$\"#?!\"\"$\"1_\\1s^g08!#:-%'LEGENDG6#-%)_TYP ESETG6#Q+Exact~y(x)6\"-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\"!!\" \"-%%VIEWG6$;$\"\"!!\"\"$\"#?!\"\";$\"1&)o_#=6g)R!#;$\"2s&pFl&*QB8!#;- %+AXESLABELSG6'Q\"x6\"Q!6\"-%%FONTG6%%!G%!G\"#5%+HORIZONTALG%+HORIZONT ALG-%&TITLEG6$-%)_TYPESETG6#QIExact~Solution~Plotted~on~Given~Interval 6\"-%-TRANSPARENCYG6#$\"\"!!\"\"-%*THICKNESSG6#\"\"#-%*AXESSTYLEG6#%'N ORMALG-%(SCALINGG6#%.UNCONSTRAINEDG" 1 2 2 1 10 2 2 6 0 4 2 1.0 45.0 45.0 0 1 "Exact y(x)" }}{TEXT 229 0 "" }}}}{PARA 259 "" 0 "" {TEXT 225 0 "" }}{SECT 0 {PARA 260 "" 0 "" {TEXT 227 31 "Section 3: Approxim ate Solution" }{TEXT 227 0 "" }}{SECT 0 {PARA 206 "" 0 "" {TEXT 243 6 "Step 1" }{TEXT 243 0 "" }}{PARA 217 "" 0 "" {TEXT 290 111 "This works heet shows how Euler's method is used over 5 steps to approximate the \+ value of the solution function " }{TEXT 255 4 "y(x)" }{TEXT 290 4 " at " }{TEXT 255 6 "x = xf" }{TEXT 290 43 ". First we find the length of \+ each segment." }{TEXT 290 0 "" }}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 15 "h:=(xf-x0)/5.0;" }{MPLTEXT 1 244 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I\"hG6\"$\"+++++S!#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 290 13 "The function " }{TEXT 255 6 "f(x,y)" }{TEXT 290 127 " is used to find the derivative at the initial condition. Thi s derivative then is used to predict the value of the function at " } {TEXT 255 15 "x[1] = x[0] + h" }{TEXT 290 51 ". The first values used \+ are the initial conditions." }{TEXT 290 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 9 "X[0]:=x0;" }{MPLTEXT 1 244 1 "\n" }{MPLTEXT 1 244 9 "Y[0]:=y0;" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>&I\"XG6\"6#\"\"! $\"\"!\"\"!" }{TEXT 20 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>&I\"YG 6\"6#\"\"!$\"#5!\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 290 9 "The next " }{TEXT 255 1 "x" }{TEXT 290 9 " value is" }{TEXT 290 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 13 "X[1]:=X[0]+h ;" }{MPLTEXT 1 244 0 "" }{MPLTEXT 1 244 1 "\n" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>&I\"XG6\"6#\"\"\"$\"+++++S!#5" }{TEXT 20 0 "" }}} {EXCHG {PARA 217 "" 0 "" {TEXT 290 82 "The derivative is evaluated at \+ the previous point (initial condition) and the new " }{TEXT 255 1 "y" }{TEXT 290 31 " is found using Euler's method." }{TEXT 290 0 "" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 245 13 "f(X[0],Y[0]);" }{MPLTEXT 1 245 0 "" }{MPLTEXT 1 245 1 "\n" }{MPLTEXT 1 245 26 "Y[1]:=Y[0]+f(X[ 0],Y[0])*h;" }{MPLTEXT 1 245 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#$ !$?\"!\"#" }{TEXT 20 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>&I\"YG6 \"6#\"\"\"$\"+++++_!#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 225 66 "For plotting we must combine the X and Y vectors into a \+ 2-D array." }{TEXT 225 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 30 "XY:=[[X[0],Y[0]],[X[1],Y[1]]]:" }{MPLTEXT 1 244 0 "" }}} {EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 159 "plot([XY,y(x)],x=x0..xf ,color=[GREEN,RED],thickness=2,legend=[\"Approximation\",\"Exact\"],ti tle=\"Exact and Approximate Solutions Using Eulers Method - 1st Step\" );" }{MPLTEXT 1 244 0 "" }}{PARA 253 "" 1 "" {TEXT 300 0 "" } {GLPLOT2D 569 449 449 {PLOTDATA 2 "6*-%'CURVESG6%7$7$$\"\"!!\"\"$\"#5! \"\"7$$\"\"%!\"\"$\"#_!\"#-%'LEGENDG6#-%)_TYPESETG6#Q.Approximation6\" -%&COLORG6&%$RGBG$\"\"!!\"\"$\"#5!\"\"$\"\"!!\"\"-%'CURVESG6%7W7$$\"\" !!\"\"$\"#5!\"\"7$$\"1LLLL3VfV!#<$\"1k$>L@y0\\*!#;7$$\"1nmm\"H[D:)!#<$ \"1:g=0Kmp!*!#;7$$\"2LLL$e0$=C\"!#<$\"1i\"\\uEB5i)!#;7$$\"2LLL$3RBr;!# <$\"1^%R.Pjb>)!#;7$$\"2lmm\"zjf)4#!#<$\"0[F(oIu(z(!#:7$$\"2KLLe4;[\\#! #<$\"1#eIj7g7X(!#;7$$\"2(****\\i'y]!H!#<$\"1x(Gtc$f9r!#;7$$\"1LL$ezs$H L!#;$\"1a4vvmR*y'!#;7$$\"2&****\\7iI_P!#<$\"0)Hhs.x(['!#:7$$\"1nmm;_M( =%!#;$\"197lkC<+i!#;7$$\"2MLL$3y_qX!#<$\"1>/U'**G_'f!#;7$$\"*l+>+&!\"* $\"1c\\`=ee?d!#;7$$\"*vW]V&!\"*$\"1u(ya*pJ&\\&!#;7$$\"*NfC&e!\"*$\"1X? MZu#oH&!#;7$$\"1LL$ez6:B'!#;$\"1p\"yvrP=8&!#;7$$\"1mmm;=C#o'!#;$\"2W1e p.,R&\\!#<7$$\"1mmmm#pS1(!#;$\"29^N!e**==[!#<7$$\",DOD#3v!#6$\"1TSN&HP rn%!#;7$$\"1mmmm(y8!z!#;$\"1K9om%zpc%!#;7$$\",DOIFL)!#6$\"0ErlzT;Y%!#: 7$$\",v3zMu)!#6$\"2$4U5t\"zhP%!#<7$$\"1nmm\"H_?<*!#;$\"2DZG*H;F-V!#<7$ $\"1nm;zihl&*!#;$\"2u47S'Q2[U!#<7$$\"1LLL3#G,***!#;$\"1bv+GA%!#<7$$\"2)***\\i&p@[7!#;$\"0'4)fEc eF%!#:7$$\"2)****\\2'HKH\"!#;$\"17F#ypikN%!#;7$$\"2lmmmZvOL\"!#;$\"1X? r,-,]W!#;7$$\"+v+'oP\"!\"*$\"1PkLBW5uX!#;7$$\"2KL$eR<*fT\"!#;$\"28@`$G DO5Z!#<7$$\"+&)Hxe9!\"*$\"1O-*Q!Q>))[!#;7$$\"2lm;H!o-*\\\"!#;$\"18\"Hx Brj3&!#;7$$\"2****\\7k.6a\"!#;$\"1\"\\+())RtH`!#;7$$\"2mmm;WTAe\"!#;$ \"1Ta:;EU3c!#;7$$\"-D1*3`i\"!#6$\"1wN`^m')\\f!#;7$$\"2MLLL*zym;!#;$\"1 PZ/Q]MMj!#;7$$\"2LLL3N1#4!#6$\"2$p4*Q6- n/\"!#;7$$\"*PDj$>!\")$\"1(*[4x+?,6!#:7$$\"-v.Uac>!#6$\"2lAmA8r.;\"!#; 7$$\".v=5s#y>!#7$\"16&>:*))pH7!#:7$$\"#?!\"\"$\"1_\\1s^g08!#:-%'LEGEND G6#-%)_TYPESETG6#Q&Exact6\"-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\" !!\"\"-%%VIEWG6$;$\"\"!!\"\"$\"#?!\"\";$\"1&)o_#=6g)R!#;$\"2s&pFl&*QB8 !#;-%+AXESLABELSG6'Q\"x6\"Q!6\"-%%FONTG6%%!G%!G\"#5%+HORIZONTALG%+HORI ZONTALG-%&TITLEG6$-%)_TYPESETG6#QinExact~and~Approximate~Solutions~Usi ng~Eulers~Method~-~1st~Step6\"-%-TRANSPARENCYG6#$\"\"!!\"\"-%*THICKNES SG6#\"\"#-%*AXESSTYLEG6#%'NORMALG-%(SCALINGG6#%.UNCONSTRAINEDG" 1 2 2 1 10 2 2 6 0 4 2 1.0 45.0 45.0 0 1 "Approximation" "Exact" }}{TEXT 229 0 "" }}}}{SECT 0 {PARA 206 "" 0 "" {TEXT 243 6 "Step 2" }{TEXT 243 0 "" }}{PARA 217 "" 0 "" {TEXT 290 45 "Now the procedure is repeat ed using the next " }{TEXT 255 1 "x" }{TEXT 290 8 " value (" }{TEXT 255 15 "x[2] = x[1] + h" }{TEXT 290 52 ") and the y value approximated in the previous step." }{TEXT 290 0 "" }}{EXCHG {PARA 217 "" 0 "" {TEXT 290 9 "The next " }{TEXT 255 1 "x" }{TEXT 290 9 " value is" } {TEXT 290 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 13 "X[2]:= X[1]+h;" }{MPLTEXT 1 244 0 "" }{MPLTEXT 1 244 1 "\n" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>&I\"XG6\"6#\"\"#$\"+++++!)!#5" }{TEXT 20 0 "" }}} {EXCHG {PARA 217 "" 0 "" {TEXT 290 62 "The derivative is evaluated at \+ the previous point and the new " }{TEXT 255 1 "y" }{TEXT 290 31 " is f ound using Euler's method." }{TEXT 290 0 "" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 245 5 "X[1];" }{MPLTEXT 1 245 1 "\n" }{MPLTEXT 1 245 5 "Y[1];" }{MPLTEXT 1 245 1 "\n" }{MPLTEXT 1 245 13 "f(X[1],Y[1]);" } {MPLTEXT 1 245 0 "" }{MPLTEXT 1 245 1 "\n" }{MPLTEXT 1 245 26 "Y[2]:=Y [1]+f(X[1],Y[1])*h;" }{MPLTEXT 1 245 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#$\"+++++S!#5" }{TEXT 20 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#$\"+++++_!#5" }{TEXT 20 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#$!++++3a!#5" }{TEXT 20 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>&I\"YG6\"6#\"\"#$\"+++!o.$!#5" }{TEXT 20 0 "" }}} {EXCHG {PARA 259 "" 0 "" {TEXT 225 66 "For plotting we must combine th e X and Y vectors into a 2-D array." }{TEXT 225 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 42 "XY:=[[X[0],Y[0]],[X[1],Y[1]],[X[2],Y[ 2]]]:" }{MPLTEXT 1 244 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 159 "plot([XY,y(x)],x=x0..xf,color=[GREEN,RED],thickness=2,legend= [\"Approximation\",\"Exact\"],title=\"Exact and Approximate Solutions \+ Using Eulers Method - 2nd Step\");" }{MPLTEXT 1 244 0 "" }}{PARA 253 " " 1 "" {TEXT 300 0 "" }{GLPLOT2D 569 449 449 {PLOTDATA 2 "6*-%'CURVESG 6%7%7$$\"\"!!\"\"$\"#5!\"\"7$$\"\"%!\"\"$\"#_!\"#7$$\"\")!\"\"$\"&o.$! \"&-%'LEGENDG6#-%)_TYPESETG6#Q.Approximation6\"-%&COLORG6&%$RGBG$\"\"! !\"\"$\"#5!\"\"$\"\"!!\"\"-%'CURVESG6%7W7$$\"\"!!\"\"$\"#5!\"\"7$$\"1L LLL3VfV!#<$\"1k$>L@y0\\*!#;7$$\"1nmm\"H[D:)!#<$\"1:g=0Kmp!*!#;7$$\"2LL L$e0$=C\"!#<$\"1i\"\\uEB5i)!#;7$$\"2LLL$3RBr;!#<$\"1^%R.Pjb>)!#;7$$\"2 lmm\"zjf)4#!#<$\"0[F(oIu(z(!#:7$$\"2KLLe4;[\\#!#<$\"1#eIj7g7X(!#;7$$\" 2(****\\i'y]!H!#<$\"1x(Gtc$f9r!#;7$$\"1LL$ezs$HL!#;$\"1a4vvmR*y'!#;7$$ \"2&****\\7iI_P!#<$\"0)Hhs.x(['!#:7$$\"1nmm;_M(=%!#;$\"197lkC<+i!#;7$$ \"2MLL$3y_qX!#<$\"1>/U'**G_'f!#;7$$\"*l+>+&!\"*$\"1c\\`=ee?d!#;7$$\"*v W]V&!\"*$\"1u(ya*pJ&\\&!#;7$$\"*NfC&e!\"*$\"1X?MZu#oH&!#;7$$\"1LL$ez6: B'!#;$\"1p\"yvrP=8&!#;7$$\"1mmm;=C#o'!#;$\"2W1ep.,R&\\!#<7$$\"1mmmm#pS 1(!#;$\"29^N!e**==[!#<7$$\",DOD#3v!#6$\"1TSN&HPrn%!#;7$$\"1mmmm(y8!z!# ;$\"1K9om%zpc%!#;7$$\",DOIFL)!#6$\"0ErlzT;Y%!#:7$$\",v3zMu)!#6$\"2$4U5 t\"zhP%!#<7$$\"1nmm\"H_?<*!#;$\"2DZG*H;F-V!#<7$$\"1nm;zihl&*!#;$\"2u47 S'Q2[U!#<7$$\"1LLL3#G,***!#;$\"1bv+GA%!#<7$$\"2)***\\i&p@[7!#;$\"0'4)fEceF%!#:7$$\"2)****\\2'HK H\"!#;$\"17F#ypikN%!#;7$$\"2lmmmZvOL\"!#;$\"1X?r,-,]W!#;7$$\"+v+'oP\"! \"*$\"1PkLBW5uX!#;7$$\"2KL$eR<*fT\"!#;$\"28@`$GDO5Z!#<7$$\"+&)Hxe9!\"* $\"1O-*Q!Q>))[!#;7$$\"2lm;H!o-*\\\"!#;$\"18\"HxBrj3&!#;7$$\"2****\\7k. 6a\"!#;$\"1\"\\+())RtH`!#;7$$\"2mmm;WTAe\"!#;$\"1Ta:;EU3c!#;7$$\"-D1*3 `i\"!#6$\"1wN`^m')\\f!#;7$$\"2MLLL*zym;!#;$\"1PZ/Q]MMj!#;7$$\"2LLL3N1# 4!#6$\"2$p4*Q6-n/\"!#;7$$\"*PDj$>!\")$ \"1(*[4x+?,6!#:7$$\"-v.Uac>!#6$\"2lAmA8r.;\"!#;7$$\".v=5s#y>!#7$\"16&> :*))pH7!#:7$$\"#?!\"\"$\"1_\\1s^g08!#:-%'LEGENDG6#-%)_TYPESETG6#Q&Exac t6\"-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\"!!\"\"-%%VIEWG6$;$\"\"! !\"\"$\"#?!\"\";$\"0,(el\\TOG!#:$\"1^i]vOkD8!#:-%+AXESLABELSG6'Q\"x6\" Q!6\"-%%FONTG6%%!G%!G\"#5%+HORIZONTALG%+HORIZONTALG-%&TITLEG6$-%)_TYPE SETG6#QinExact~and~Approximate~Solutions~Using~Eulers~Method~-~2nd~Ste p6\"-%-TRANSPARENCYG6#$\"\"!!\"\"-%*THICKNESSG6#\"\"#-%*AXESSTYLEG6#%' NORMALG-%(SCALINGG6#%.UNCONSTRAINEDG" 1 2 2 1 10 2 2 6 0 4 2 1.0 45.0 45.0 0 1 "Approximation" "Exact" }}{TEXT 229 0 "" }}}}{SECT 0 {PARA 206 "" 0 "" {TEXT 243 6 "Step 3" }{TEXT 243 0 "" }}{PARA 217 "" 0 "" {TEXT 290 41 "The procedure is repeated using the next " }{TEXT 255 1 "x" }{TEXT 290 8 " value (" }{TEXT 255 15 "x[3] = x[2] + h" }{TEXT 290 52 ") and the y value approximated in the previous step." }{TEXT 290 0 "" }}{EXCHG {PARA 217 "" 0 "" {TEXT 290 9 "The next " }{TEXT 255 1 "x" }{TEXT 290 9 " value is" }{TEXT 290 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 13 "X[3]:=X[2]+h;" }{MPLTEXT 1 244 0 "" } {MPLTEXT 1 244 1 "\n" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>&I\"XG6\"6# \"\"$$\"+++++7!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 290 62 "The derivative is evaluated at the previous point and the new " }{TEXT 255 1 "y" }{TEXT 290 31 " is found using Euler's method." } {TEXT 290 0 "" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 245 5 "X[2];" }{MPLTEXT 1 245 1 "\n" }{MPLTEXT 1 245 5 "Y[2];" }{MPLTEXT 1 245 1 "\n " }{MPLTEXT 1 245 13 "f(X[2],Y[2]);" }{MPLTEXT 1 245 0 "" }{MPLTEXT 1 245 1 "\n" }{MPLTEXT 1 245 26 "Y[3]:=Y[2]+f(X[2],Y[2])*h;" }{MPLTEXT 1 245 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#$\"+++++!)!#5" }{TEXT 20 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#$\"+++!o.$!#5" }{TEXT 20 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#$!++!31q\"!#5" }{TEXT 20 0 "" }} {PARA 221 "" 1 "" {XPPMATH 20 "6#>&I\"YG6\"6#\"\"$$\"++obcB!#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 225 66 "For plotting we mus t combine the X and Y vectors into a 2-D array." }{TEXT 225 0 "" }}} {EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 54 "XY:=[[X[0],Y[0]],[X[1],Y [1]],[X[2],Y[2]],[X[3],Y[3]]]:" }{MPLTEXT 1 244 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 159 "plot([XY,y(x)],x=x0..xf,color=[GREEN ,RED],thickness=2,legend=[\"Approximation\",\"Exact\"],title=\"Exact a nd Approximate Solutions Using Eulers Method - 3rd Step\");" }{MPLTEXT 1 244 0 "" }}{PARA 253 "" 1 "" {TEXT 300 0 "" }{GLPLOT2D 613 613 613 {PLOTDATA 2 "6*-%'CURVESG6%7&7$$\"\"!!\"\"$\"#5!\"\"7$$\"\"%!\"\"$\"#_ !\"#7$$\"\")!\"\"$\"&o.$!\"&7$$\"#7!\"\"$\")obcB!\")-%'LEGENDG6#-%)_TY PESETG6#Q.Approximation6\"-%&COLORG6&%$RGBG$\"\"!!\"\"$\"#5!\"\"$\"\"! !\"\"-%'CURVESG6%7W7$$\"\"!!\"\"$\"#5!\"\"7$$\"1LLLL3VfV!#<$\"1k$>L@y0 \\*!#;7$$\"1nmm\"H[D:)!#<$\"1:g=0Kmp!*!#;7$$\"2LLL$e0$=C\"!#<$\"1i\" \\uEB5i)!#;7$$\"2LLL$3RBr;!#<$\"1^%R.Pjb>)!#;7$$\"2lmm\"zjf)4#!#<$\"0[ F(oIu(z(!#:7$$\"2KLLe4;[\\#!#<$\"1#eIj7g7X(!#;7$$\"2(****\\i'y]!H!#<$ \"1x(Gtc$f9r!#;7$$\"1LL$ezs$HL!#;$\"1a4vvmR*y'!#;7$$\"2&****\\7iI_P!#< $\"0)Hhs.x(['!#:7$$\"1nmm;_M(=%!#;$\"197lkC<+i!#;7$$\"2MLL$3y_qX!#<$\" 1>/U'**G_'f!#;7$$\"*l+>+&!\"*$\"1c\\`=ee?d!#;7$$\"*vW]V&!\"*$\"1u(ya*p J&\\&!#;7$$\"*NfC&e!\"*$\"1X?MZu#oH&!#;7$$\"1LL$ez6:B'!#;$\"1p\"yvrP=8 &!#;7$$\"1mmm;=C#o'!#;$\"2W1ep.,R&\\!#<7$$\"1mmmm#pS1(!#;$\"29^N!e**== [!#<7$$\",DOD#3v!#6$\"1TSN&HPrn%!#;7$$\"1mmmm(y8!z!#;$\"1K9om%zpc%!#;7 $$\",DOIFL)!#6$\"0ErlzT;Y%!#:7$$\",v3zMu)!#6$\"2$4U5t\"zhP%!#<7$$\"1nm m\"H_?<*!#;$\"2DZG*H;F-V!#<7$$\"1nm;zihl&*!#;$\"2u47S'Q2[U!#<7$$\"1LLL 3#G,***!#;$\"1bv+GA%!#<7$ $\"2)***\\i&p@[7!#;$\"0'4)fEceF%!#:7$$\"2)****\\2'HKH\"!#;$\"17F#ypikN %!#;7$$\"2lmmmZvOL\"!#;$\"1X?r,-,]W!#;7$$\"+v+'oP\"!\"*$\"1PkLBW5uX!#; 7$$\"2KL$eR<*fT\"!#;$\"28@`$GDO5Z!#<7$$\"+&)Hxe9!\"*$\"1O-*Q!Q>))[!#;7 $$\"2lm;H!o-*\\\"!#;$\"18\"HxBrj3&!#;7$$\"2****\\7k.6a\"!#;$\"1\"\\+() )RtH`!#;7$$\"2mmm;WTAe\"!#;$\"1Ta:;EU3c!#;7$$\"-D1*3`i\"!#6$\"1wN`^m') \\f!#;7$$\"2MLLL*zym;!#;$\"1PZ/Q]MMj!#;7$$\"2LLL3N1#4!#6$\"2$p4*Q6-n/\"!#;7$$\"*PDj$>!\")$\"1(*[4x+?,6!#:7$$ \"-v.Uac>!#6$\"2lAmA8r.;\"!#;7$$\".v=5s#y>!#7$\"16&>:*))pH7!#:7$$\"#?! \"\"$\"1_\\1s^g08!#:-%'LEGENDG6#-%)_TYPESETG6#Q&Exact6\"-%&COLORG6&%$R GBG$\"#5!\"\"$\"\"!!\"\"$\"\"!!\"\"-%%VIEWG6$;$\"\"!!\"\"$\"#?!\"\";$ \"0,(e,pcU@!#:$\"1^i!>;/qK\"!#:-%+AXESLABELSG6'Q\"x6\"Q!6\"-%%FONTG6%% !G%!G\"#5%+HORIZONTALG%+HORIZONTALG-%&TITLEG6$-%)_TYPESETG6#QinExact~a nd~Approximate~Solutions~Using~Eulers~Method~-~3rd~Step6\"-%-TRANSPARE NCYG6#$\"\"!!\"\"-%*THICKNESSG6#\"\"#-%*AXESSTYLEG6#%'NORMALG-%(SCALIN GG6#%.UNCONSTRAINEDG" 1 2 2 1 10 2 2 6 0 4 2 1.0 45.0 45.0 0 1 "Approx imation" "Exact" }}{TEXT 229 0 "" }}}}{SECT 0 {PARA 206 "" 0 "" {TEXT 243 6 "Step 4" }{TEXT 243 0 "" }}{PARA 217 "" 0 "" {TEXT 290 41 "The p rocedure is repeated using the next " }{TEXT 255 1 "x" }{TEXT 290 8 " \+ value (" }{TEXT 255 15 "x[4] = x[3] + h" }{TEXT 290 52 ") and the y va lue approximated in the previous step." }{TEXT 290 0 "" }}{EXCHG {PARA 217 "" 0 "" {TEXT 290 9 "The next " }{TEXT 255 1 "x" }{TEXT 290 9 " value is" }{TEXT 290 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 13 "X[4]:=X[3]+h;" }{MPLTEXT 1 244 0 "" }{MPLTEXT 1 244 1 "\n" } }{PARA 221 "" 1 "" {XPPMATH 20 "6#>&I\"XG6\"6#\"\"%$\"+++++;!\"*" } {TEXT 20 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 290 62 "The derivative is evaluated at the previous point and the new " }{TEXT 255 1 "y" } {TEXT 290 31 " is found using Euler's method." }{TEXT 290 0 "" }}} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 245 5 "X[3];" }{MPLTEXT 1 245 1 "\n" }{MPLTEXT 1 245 5 "Y[3];" }{MPLTEXT 1 245 1 "\n" }{MPLTEXT 1 245 13 "f(X[3],Y[3]);" }{MPLTEXT 1 245 0 "" }{MPLTEXT 1 245 1 "\n" } {MPLTEXT 1 245 26 "Y[4]:=Y[3]+f(X[3],Y[3])*h;" }{MPLTEXT 1 245 0 "" }} {PARA 221 "" 1 "" {XPPMATH 20 "6#$\"+++++7!\"*" }{TEXT 20 0 "" }} {PARA 221 "" 1 "" {XPPMATH 20 "6#$\"++obcB!#5" }{TEXT 20 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#$\"*KOdl&!#5" }{TEXT 20 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>&I\"YG6\"6#\"\"%$\"+`iy#e#!#5" }{TEXT 20 0 "" }} }{EXCHG {PARA 259 "" 0 "" {TEXT 225 66 "For plotting we must combine t he X and Y vectors into a 2-D array." }{TEXT 225 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 66 "XY:=[[X[0],Y[0]],[X[1],Y[1]],[X[2],Y[ 2]],[X[3],Y[3]],[X[4],Y[4]]]:" }{MPLTEXT 1 244 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 159 "plot([XY,y(x)],x=x0..xf,color=[GREEN ,RED],thickness=2,legend=[\"Approximation\",\"Exact\"],title=\"Exact a nd Approximate Solutions Using Eulers Method - 4th Step\");" }{MPLTEXT 1 244 0 "" }}{PARA 253 "" 1 "" {TEXT 300 0 "" }{GLPLOT2D 569 449 449 {PLOTDATA 2 "6*-%'CURVESG6%7'7$$\"\"!!\"\"$\"#5!\"\"7$$\"\"%!\"\"$\"#_ !\"#7$$\"\")!\"\"$\"&o.$!\"&7$$\"#7!\"\"$\")obcB!\")7$$\"#;!\"\"$\"+`i y#e#!#5-%'LEGENDG6#-%)_TYPESETG6#Q.Approximation6\"-%&COLORG6&%$RGBG$ \"\"!!\"\"$\"#5!\"\"$\"\"!!\"\"-%'CURVESG6%7W7$$\"\"!!\"\"$\"#5!\"\"7$ $\"1LLLL3VfV!#<$\"1k$>L@y0\\*!#;7$$\"1nmm\"H[D:)!#<$\"1:g=0Kmp!*!#;7$$ \"2LLL$e0$=C\"!#<$\"1i\"\\uEB5i)!#;7$$\"2LLL$3RBr;!#<$\"1^%R.Pjb>)!#;7 $$\"2lmm\"zjf)4#!#<$\"0[F(oIu(z(!#:7$$\"2KLLe4;[\\#!#<$\"1#eIj7g7X(!#; 7$$\"2(****\\i'y]!H!#<$\"1x(Gtc$f9r!#;7$$\"1LL$ezs$HL!#;$\"1a4vvmR*y'! #;7$$\"2&****\\7iI_P!#<$\"0)Hhs.x(['!#:7$$\"1nmm;_M(=%!#;$\"197lkC<+i! #;7$$\"2MLL$3y_qX!#<$\"1>/U'**G_'f!#;7$$\"*l+>+&!\"*$\"1c\\`=ee?d!#;7$ $\"*vW]V&!\"*$\"1u(ya*pJ&\\&!#;7$$\"*NfC&e!\"*$\"1X?MZu#oH&!#;7$$\"1LL $ez6:B'!#;$\"1p\"yvrP=8&!#;7$$\"1mmm;=C#o'!#;$\"2W1ep.,R&\\!#<7$$\"1mm mm#pS1(!#;$\"29^N!e**==[!#<7$$\",DOD#3v!#6$\"1TSN&HPrn%!#;7$$\"1mmmm(y 8!z!#;$\"1K9om%zpc%!#;7$$\",DOIFL)!#6$\"0ErlzT;Y%!#:7$$\",v3zMu)!#6$\" 2$4U5t\"zhP%!#<7$$\"1nmm\"H_?<*!#;$\"2DZG*H;F-V!#<7$$\"1nm;zihl&*!#;$ \"2u47S'Q2[U!#<7$$\"1LLL3#G,***!#;$\"1bv+GA%!#<7$$\"2)***\\i&p@[7!#;$\"0'4)fEceF%!#:7$$\"2)**** \\2'HKH\"!#;$\"17F#ypikN%!#;7$$\"2lmmmZvOL\"!#;$\"1X?r,-,]W!#;7$$\"+v+ 'oP\"!\"*$\"1PkLBW5uX!#;7$$\"2KL$eR<*fT\"!#;$\"28@`$GDO5Z!#<7$$\"+&)Hx e9!\"*$\"1O-*Q!Q>))[!#;7$$\"2lm;H!o-*\\\"!#;$\"18\"HxBrj3&!#;7$$\"2*** *\\7k.6a\"!#;$\"1\"\\+())RtH`!#;7$$\"2mmm;WTAe\"!#;$\"1Ta:;EU3c!#;7$$ \"-D1*3`i\"!#6$\"1wN`^m')\\f!#;7$$\"2MLLL*zym;!#;$\"1PZ/Q]MMj!#;7$$\"2 LLL3N1#4!#6$\"2$p4*Q6-n/\"!#;7$$\"*PDj$ >!\")$\"1(*[4x+?,6!#:7$$\"-v.Uac>!#6$\"2lAmA8r.;\"!#;7$$\".v=5s#y>!#7$ \"16&>:*))pH7!#:7$$\"#?!\"\"$\"1_\\1s^g08!#:-%'LEGENDG6#-%)_TYPESETG6# Q&Exact6\"-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\"!!\"\"-%%VIEWG6$; $\"\"!!\"\"$\"#?!\"\";$\"0,(e,pcU@!#:$\"1^i!>;/qK\"!#:-%+AXESLABELSG6' Q\"x6\"Q!6\"-%%FONTG6%%!G%!G\"#5%+HORIZONTALG%+HORIZONTALG-%&TITLEG6$- %)_TYPESETG6#QinExact~and~Approximate~Solutions~Using~Eulers~Method~-~ 4th~Step6\"-%-TRANSPARENCYG6#$\"\"!!\"\"-%*THICKNESSG6#\"\"#-%*AXESSTY LEG6#%'NORMALG-%(SCALINGG6#%.UNCONSTRAINEDG" 1 2 2 1 10 2 2 6 0 4 2 1.0 45.0 45.0 0 1 "Approximation" "Exact" }}{TEXT 229 0 "" }}}}{SECT 0 {PARA 206 "" 0 "" {TEXT 243 6 "Step 5" }{TEXT 243 0 "" }}{PARA 217 "" 0 "" {TEXT 290 41 "The procedure is repeated using the next " }{TEXT 255 1 "x" }{TEXT 290 8 " value (" }{TEXT 255 15 "x[5] = x[4] + h" } {TEXT 290 52 ") and the y value approximated in the previous step." } {TEXT 290 0 "" }}{EXCHG {PARA 217 "" 0 "" {TEXT 290 9 "The next " } {TEXT 255 1 "x" }{TEXT 290 9 " value is" }{TEXT 290 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 13 "X[5]:=X[4]+h;" }{MPLTEXT 1 244 0 "" }{MPLTEXT 1 244 1 "\n" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>&I\"XG 6\"6#\"\"&$\"+++++?!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 217 "" 0 "" {TEXT 290 62 "The derivative is evaluated at the previous point and th e new " }{TEXT 255 1 "y" }{TEXT 290 31 " is found using Euler's method ." }{TEXT 290 0 "" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 245 5 "X[4 ];" }{MPLTEXT 1 245 1 "\n" }{MPLTEXT 1 245 5 "Y[4];" }{MPLTEXT 1 245 1 "\n" }{MPLTEXT 1 245 13 "f(X[4],Y[4]);" }{MPLTEXT 1 245 0 "" } {MPLTEXT 1 245 1 "\n" }{MPLTEXT 1 245 26 "Y[5]:=Y[4]+f(X[4],Y[4])*h;" }{MPLTEXT 1 245 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#$\"+++++;!\"*" }{TEXT 20 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#$\"+`iy#e#!#5" } {TEXT 20 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#$\"+/$*e7N!#5" }{TEXT 20 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>&I\"YG6\"6#\"\"&$\"+v>#y) R!#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 225 66 "For plo tting we must combine the X and Y vectors into a 2-D array." }{TEXT 225 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 78 "XY:=[[X[0],Y [0]],[X[1],Y[1]],[X[2],Y[2]],[X[3],Y[3]],[X[4],Y[4]],[X[5],Y[5]]]:" } {MPLTEXT 1 244 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 159 " plot([XY,y(x)],x=x0..xf,color=[GREEN,RED],thickness=2,legend=[\"Approx imation\",\"Exact\"],title=\"Exact and Approximate Solutions Using Eul ers Method - 5th Step\");" }{MPLTEXT 1 244 0 "" }}{PARA 253 "" 1 "" {TEXT 300 0 "" }{GLPLOT2D 599 447 447 {PLOTDATA 2 "6*-%'CURVESG6%7(7$$ \"\"!!\"\"$\"#5!\"\"7$$\"\"%!\"\"$\"#_!\"#7$$\"\")!\"\"$\"&o.$!\"&7$$ \"#7!\"\"$\")obcB!\")7$$\"#;!\"\"$\"+`iy#e#!#57$$\"#?!\"\"$\"+v>#y)R!# 5-%'LEGENDG6#-%)_TYPESETG6#Q.Approximation6\"-%&COLORG6&%$RGBG$\"\"!! \"\"$\"#5!\"\"$\"\"!!\"\"-%'CURVESG6%7W7$$\"\"!!\"\"$\"#5!\"\"7$$\"1LL LL3VfV!#<$\"1k$>L@y0\\*!#;7$$\"1nmm\"H[D:)!#<$\"1:g=0Kmp!*!#;7$$\"2LLL $e0$=C\"!#<$\"1i\"\\uEB5i)!#;7$$\"2LLL$3RBr;!#<$\"1^%R.Pjb>)!#;7$$\"2l mm\"zjf)4#!#<$\"0[F(oIu(z(!#:7$$\"2KLLe4;[\\#!#<$\"1#eIj7g7X(!#;7$$\"2 (****\\i'y]!H!#<$\"1x(Gtc$f9r!#;7$$\"1LL$ezs$HL!#;$\"1a4vvmR*y'!#;7$$ \"2&****\\7iI_P!#<$\"0)Hhs.x(['!#:7$$\"1nmm;_M(=%!#;$\"197lkC<+i!#;7$$ \"2MLL$3y_qX!#<$\"1>/U'**G_'f!#;7$$\"*l+>+&!\"*$\"1c\\`=ee?d!#;7$$\"*v W]V&!\"*$\"1u(ya*pJ&\\&!#;7$$\"*NfC&e!\"*$\"1X?MZu#oH&!#;7$$\"1LL$ez6: B'!#;$\"1p\"yvrP=8&!#;7$$\"1mmm;=C#o'!#;$\"2W1ep.,R&\\!#<7$$\"1mmmm#pS 1(!#;$\"29^N!e**==[!#<7$$\",DOD#3v!#6$\"1TSN&HPrn%!#;7$$\"1mmmm(y8!z!# ;$\"1K9om%zpc%!#;7$$\",DOIFL)!#6$\"0ErlzT;Y%!#:7$$\",v3zMu)!#6$\"2$4U5 t\"zhP%!#<7$$\"1nmm\"H_?<*!#;$\"2DZG*H;F-V!#<7$$\"1nm;zihl&*!#;$\"2u47 S'Q2[U!#<7$$\"1LLL3#G,***!#;$\"1bv+GA%!#<7$$\"2)***\\i&p@[7!#;$\"0'4)fEceF%!#:7$$\"2)****\\2'HK H\"!#;$\"17F#ypikN%!#;7$$\"2lmmmZvOL\"!#;$\"1X?r,-,]W!#;7$$\"+v+'oP\"! \"*$\"1PkLBW5uX!#;7$$\"2KL$eR<*fT\"!#;$\"28@`$GDO5Z!#<7$$\"+&)Hxe9!\"* $\"1O-*Q!Q>))[!#;7$$\"2lm;H!o-*\\\"!#;$\"18\"HxBrj3&!#;7$$\"2****\\7k. 6a\"!#;$\"1\"\\+())RtH`!#;7$$\"2mmm;WTAe\"!#;$\"1Ta:;EU3c!#;7$$\"-D1*3 `i\"!#6$\"1wN`^m')\\f!#;7$$\"2MLLL*zym;!#;$\"1PZ/Q]MMj!#;7$$\"2LLL3N1# 4!#6$\"2$p4*Q6-n/\"!#;7$$\"*PDj$>!\")$ \"1(*[4x+?,6!#:7$$\"-v.Uac>!#6$\"2lAmA8r.;\"!#;7$$\".v=5s#y>!#7$\"16&> :*))pH7!#:7$$\"#?!\"\"$\"1_\\1s^g08!#:-%'LEGENDG6#-%)_TYPESETG6#Q&Exac t6\"-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\"!!\"\"-%%VIEWG6$;$\"\"! !\"\"$\"#?!\"\";$\"0,(e,pcU@!#:$\"1^i!>;/qK\"!#:-%+AXESLABELSG6'Q\"x6 \"Q!6\"-%%FONTG6%%!G%!G\"#5%+HORIZONTALG%+HORIZONTALG-%&TITLEG6$-%)_TY PESETG6#QinExact~and~Approximate~Solutions~Using~Eulers~Method~-~5th~S tep6\"-%-TRANSPARENCYG6#$\"\"!!\"\"-%*THICKNESSG6#\"\"#-%*AXESSTYLEG6# %'NORMALG-%(SCALINGG6#%.UNCONSTRAINEDG" 1 2 2 1 10 2 2 6 0 4 2 1.0 45.0 45.0 0 1 "Approximation" "Exact" }}{TEXT 229 0 "" }}}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 227 10 "References" }{TEXT 227 0 "" }}{EXCHG {PARA 217 "" 0 "" {TEXT 290 17 "Chapra & Canale, " }{TEXT 255 64 "Nume rical Methods for Engineering: With Programming Applications" }{TEXT 290 32 ". 3rd ed. WCB/McGraw-Hill. 1998." }{TEXT 290 0 "" }}}}{SECT 0 {PARA 244 "" 0 "" {TEXT 269 11 "Conclusions" }{TEXT 269 0 "" }}{PARA 217 "" 0 "" {TEXT 290 283 "While Euler's Method is valid for approxima ting the solutions of ordinary differential equations, the use of the \+ slope at one point to project the value at the next point is not very \+ accurate. Note the values obtained as well as the true and relative tr ue error at our desired point " }{TEXT 255 4 "x=xf" }{TEXT 290 1 "." } {TEXT 290 0 "" }}{EXCHG {PARA 259 "" 0 "" {TEXT 225 11 "Exact Value" } {TEXT 225 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 27 "yf:=ev alf(subs(x=xf,y(x)));" }{MPLTEXT 1 244 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I#yfG6\"$\"+s^g08!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 225 17 "Approximate Value" }{TEXT 225 0 "" }}} {EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 5 "Y[5];" }{MPLTEXT 1 244 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#$\"+v>#y)R!#5" }{TEXT 20 0 "" }} }{EXCHG {PARA 259 "" 0 "" {TEXT 225 10 "True Error" }{TEXT 225 0 "" }} }{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 32 "Et:=Y[5]-evalf(subs(x=x f,y(x)));" }{MPLTEXT 1 244 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I# EtG6\"$!+X(H#o!*!#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 225 39 "Absolute Relative True Error Percentage" }{TEXT 225 0 "" }}} {EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 244 39 "et:=abs(Et/evalf(subs(x= xf,y(x)))*100);" }{MPLTEXT 1 244 0 "" }}{PARA 221 "" 1 "" {XPPMATH 20 "6#>I#etG6\"$\"+TThXp!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 225 329 "That is a lot of error! Here, our approximation would \+ be better if we made our segments smaller and more in number. Other me thods use the same principle as Euler's method: use some approximation of slope to find the value at a point a fixed distance away. The diff erence in methods is the function used to approximate the slope." } {TEXT 225 0 "" }}}}{PARA 220 "" 0 "" {TEXT 221 0 "" }}{PARA 220 "" 0 " " {TEXT 221 398 "Legal Notice: The copyright for this application is o wned by the author(s). Neither Maplesoft nor the author are responsibl e for any errors contained within and are not liable for any damages r esulting from the use of this material. This application is intended f or non-commercial, non-profit use only. Contact the author for permiss ion if you wish to use this application in for-profit activities." } {TEXT 221 0 "" }}{PARA 236 "" 0 "" {TEXT 295 0 "" }}{PARA 229 "" 0 "" {TEXT 286 0 "" }}{PARA 256 "" 0 "" {TEXT 301 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }