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0 0 0 0 1 }{CSTYLE "" -1 324 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 325 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 326 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 327 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 328 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 329 "" 1 14 0 0 0 0 0 0 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 "Heading 1 " 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 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 "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 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 "" 18 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 19 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT -1 16 "Series Solutions " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 329 30 "Differential Equati ons Project" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" } {TEXT 257 0 "" }{TEXT 258 11 "Objectives:" }}{PARA 0 "" 0 "" {TEXT -1 98 "(a) To understand the use of power series in approximating a sol ution to a differential equation" }}{PARA 0 "" 0 "" {TEXT -1 69 " \+ whose coefficients are functions of the independent variable. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "(b) To \+ introduce the " }{TEXT 276 6 "series" }{TEXT -1 16 " option of the " }{TEXT 259 6 "dsolve" }{TEXT -1 35 " command, as well as the commands " }{TEXT 325 0 "" }{TEXT 278 0 "" }{TEXT 277 5 "Order" }}{PARA 0 "" 0 "" {TEXT -1 12 " and " }{TEXT 279 0 "" }{TEXT 280 0 "" } {TEXT 326 7 "convert" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 88 " The power series metho d will allow us to handle differential equations with " }{TEXT 327 8 " variable" }}{PARA 0 "" 0 "" {TEXT -1 102 " coefficients, most of which cannot be solved in closed form; but this method involves quit e a " }}{PARA 0 "" 0 "" {TEXT -1 27 " bit of computation. " } {TEXT 282 7 " Maple " }{TEXT -1 12 "has built-in" }{TEXT 281 1 " " } {TEXT -1 51 "capabilities that ease the computational burden and" }} {PARA 0 "" 0 "" {TEXT -1 68 " allow us to focus on the meaning o f the approximate solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 260 17 "Solved Example 1:" }} {PARA 0 "" 0 "" {TEXT -1 33 "Solve the initial value problem " } {TEXT 283 1 "y" }{TEXT -1 7 "'' - " }{TEXT 284 1 "x" }{TEXT -1 1 " \+ " }{TEXT 285 1 "y" }{TEXT -1 8 "' + 4 " }{TEXT 286 1 "y" }{TEXT -1 5 " ; " }{TEXT 287 1 "y" }{TEXT -1 10 "(0) = 1, " }{TEXT 288 2 " y " }{TEXT -1 9 "'(0) = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 261 0 "" }{TEXT 262 9 "Solution: " }}{PARA 0 "" 0 "" {TEXT -1 87 "First we note that the equation does \+ not have constant coefficients. Next we see that " }{TEXT 299 1 "x" } {TEXT -1 12 " = 0 is an " }{TEXT 300 14 "ordinary point" }{TEXT -1 246 " of the equation. Existence and uniqueness theory guarantees tha t this IVP has a unique solution. The theory of series solutions also guarantees that the solution has a Maclaurin series expansion that re presents the solution at every value of " }{TEXT 289 1 "x" }{TEXT -1 84 ". To control the size of the degree of the polynomial approximati on, we invoke the " }{TEXT 296 0 "" }{TEXT 297 5 "Order" }{TEXT 298 0 "" }{TEXT -1 109 " command, first asking to see all the terms of the s eries up to order 3 and then up to order 10. We use the " }{TEXT 290 0 "" }{TEXT 291 6 "series" }{TEXT 292 0 "" }{TEXT -1 11 " option in " }{TEXT 293 0 "" }{TEXT 294 6 "dsolve" }{TEXT 295 0 "" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Ord er:=3:\nde:=diff(y(x),x$2)-x*diff(y(x),x)+4*y(x);\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#\"\"\"* &F,F1-F'6$F)F,F1!\"\"*&\"\"%F1F)F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dsolve(\{de,y(0)=1,D(y)(0)=0\}, y(x), type=series);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG+)F'\"\"\"\"\"!!\"# \"\"#-%\"OG6#F)\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Ord er:=10:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dsolve(\{de,y(0) =1,D(y)(0)=0\}, y(x), type=series);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(\"\"\"F)*&\"\"#F))F'F+F)!\"\"*&#F)\"\"$F)*$)F'\"\"% F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Order:=20:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dsolve(\{de,y(0)=1,D(y)(0)=0\}, y(x), type=series);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(\"\"\"F)*&\"\"#F)) F'F+F)!\"\"*&#F)\"\"$F)*$)F'\"\"%F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "It seems as if the Maclaurin series for the solution is j ust the polynomial " }{XPPEDIT 18 0 "1-2*x^2+x^4/3;" "6#,(\"\"\"F$*& \"\"#F$*$%\"xGF&F$!\"\"*&F(\"\"%\"\"$F)F$" }{TEXT -1 141 " . We can c heck this and see that we have been lucky enough to find a simple clos ed form solution in the form of a fourth degree polynomial." }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 263 17 "Solved Example 2: " }}{PARA 0 "" 0 "" {TEXT -1 20 "Solve the equation " }{XPPEDIT 18 0 "(5*x^2-2)*diff(y,`$`(x,2))+15*x*diff(y,x)+5*y = 0;" "6#/,(*&,&*&\"\"& \"\"\"*$%\"xG\"\"#F)F)F,!\"\"F)-%%diffG6$%\"yG-%\"$G6$F+F,F)F)*(\"#:F) F+F)-F/6$F1F+F)F)*&F(F)F1F)F)\"\"!" }{TEXT -1 18 " near the point " }{TEXT 301 1 "x" }{TEXT -1 5 " = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 264 0 "" }{TEXT 265 0 " " }{TEXT 266 0 "" }{TEXT 267 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 30 "The only singular points are " }{XPPEDIT 18 0 "x = sqrt(2/5);" "6 #/%\"xG-%%sqrtG6#*&\"\"#\"\"\"\"\"&!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "x = -sqrt(2/5);" "6#/%\"xG,$-%%sqrtG6#*&\"\"#\"\"\"\"\" &!\"\"F-" }{TEXT -1 12 " , so that " }{TEXT 302 1 "x" }{TEXT -1 151 " = 0 is an ordinary point. Therefore we can represent the general sol ution of our equation as a Maclaurin series with radius of convergence at least " }{XPPEDIT 18 0 "sqrt(2/5)" "6#-%%sqrtG6#*&\"\"#\"\"\"\"\" &!\"\"" }{TEXT -1 58 " . Let's get an idea of the form of the series solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "Order:=10:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "de2:=(5*x^2-2)*diff(y(x),x$2)+15*x*diff(y(x),x)+5*y(x)=0;\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de2G/,(*&,&*&\"\"&\"\"\")%\"xG\"\"# F+F+F.!\"\"F+-%%diffG6$-%\"yG6#F--%\"$G6$F-F.F+F+*(\"#:F+F-F+-F16$F3F- F+F+*&F*F+F3F+F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ds olve(de2,y(x),type=series);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\" yG6#%\"xG+9F'-F%6#\"\"!F+--%\"DG6#F%F*\"\"\",$*&#\"\"&\"\"%F0F)F0F0\" \"#,$*&#F4\"\"$F0F,F0F0F:,$*&#\"#v\"#KF0F)F0F0F5,$*&#\"#5F:F0F,F0F0F4, $*&#\"$D'\"$G\"F0F)F0F0\"\"',$*&#\"#]\"\"(F0F,F0F0FN,$*&#\"&v=#\"%[?F0 F)F0F0\"\"),$*&#\"%+5\"#jF0F,F0F0\"\"*-%\"OG6#F0FC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "convert(rhs(%), polynom);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,6- %\"yG6#\"\"!\"\"\"*&%\"xGF(--%\"DG6#F%F&F(F(*&#\"\"&\"\"%F(*&F$F()F*\" \"#F(F(F(*&#F1\"\"$F(*&F+F()F*F8F(F(F(*&#\"#v\"#KF(*&F$F()F*F2F(F(F(*& #\"#5F8F(*&F+F()F*F1F(F(F(*&#\"$D'\"$G\"F(*&F$F()F*\"\"'F(F(F(*&#\"#] \"\"(F(*&F+F()F*FPF(F(F(*&#\"&v=#\"%[?F(*&F$F()F*\"\")F(F(F(*&#\"%+5\" #jF(*&F+F()F*\"\"*F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 303 17 "Solved Example 3:" }} {PARA 0 "" 0 "" {TEXT -1 53 "Use a series method to find the general s olution of " }{XPPEDIT 18 0 "2*x*diff(y,`$`(x,2))+(1-2*x^2)*diff(y,x) -4*x*y = 0;" "6#/,(*(\"\"#\"\"\"%\"xGF'-%%diffG6$%\"yG-%\"$G6$F(F&F'F' *&,&F'F'*&F&F'*$F(F&F'!\"\"F'-F*6$F,F(F'F'*(\"\"%F'F(F'F,F'F4\"\"!" } {TEXT -1 3 ", " }{TEXT 304 1 "x" }{TEXT -1 22 " > 0, near the point \+ " }{TEXT 305 1 "x" }{TEXT -1 5 " = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 306 0 "" }{TEXT 307 8 "Solution" }{TEXT 308 0 "" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "First we note that " }{TEXT 309 1 "x" }{TEXT -1 11 " = 0 is a " }{TEXT 328 7 "regular" }{TEXT -1 1 " " }{TEXT 310 14 "singular point" }{TEXT -1 98 ". Although a regul ar power series will not work, we could try to find solutions in the f orm of a " }{TEXT 311 19 "Frobenius series " }{XPPEDIT 312 0 "y(x) = sum(c[n]*x^(n+r),n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$sumG6$*&&% \"cG6#%\"nG\"\"\")F',&F/F0%\"rGF0F0/F/;\"\"!%)infinityG" }{TEXT -1 21 ", where the number " }{TEXT 313 1 "r" }{TEXT -1 22 " satifies a ce rtain " }{TEXT 314 26 "indicial equation. Maple " }{TEXT -1 27 "hand les all this with ease:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 58 "DE:=2*x*diff(y(x),x$2)+(1-2*x^2)*diff(y(x),x )-4*x*y(x)=0;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/,(*(\"\"#\" \"\"%\"xGF)-%%diffG6$-%\"yG6#F*-%\"$G6$F*F(F)F)*&,&F)F)*&F(F))F*F(F)! \"\"F)-F,6$F.F*F)F)*(\"\"%F)F*F)F.F)F8\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dsolve(DE,y(x),type=series);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(%$_C1G\"\"\"F'#F+\"\"#+/F'F+\"\"!F,F- #F+\"\")\"\"%#F+\"#[\"\"'#F+\"$%QF1-%\"OG6#F+\"#5F+F+*&%$_C2GF++/F'F+F /#F-\"\"$F-#F2\"#@F2#F1\"$J#F5#\"#;\"%lMF1F8F;F+F+" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 96 "______________________ ______________________________________________________________________ ____" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 268 0 "" }{TEXT 269 10 "ASSIGNMENT" }{TEXT -1 0 "" }{TEXT 275 0 "" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 270 10 "Problem 1:" }}{PARA 0 "" 0 "" {TEXT -1 61 "Find the power seri es solution of the initial value problem " }}{PARA 0 "" 0 "" {TEXT -1 28 " " }{TEXT 315 1 "y" }{TEXT -1 7 "'' \+ - " }{TEXT 316 2 "x " }{TEXT -1 0 "" }{TEXT 317 1 "y" }{TEXT -1 6 "' - " }{TEXT 318 1 "y" }{TEXT -1 10 " = 0 ; " }{TEXT 319 1 "y" } {TEXT -1 10 "(0) = 2, " }{TEXT 320 1 "y" }{TEXT -1 9 "'(0) = 1," }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 271 0 "" }{TEXT 272 0 "" }{TEXT 273 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 32 "Find the general solution near " }{TEXT 321 1 "x" }{TEXT -1 11 " = 0 of " }{XPPEDIT 18 0 "(4-x^2)*diff(y,`$`(x,2) )-2*x*diff(y,x)+2*y = 0;" "6#/,(*&,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"F(-%%d iffG6$%\"yG-%\"$G6$F*F+F(F(*(F+F(F*F(-F.6$F0F*F(F,*&F+F(F0F(F(\"\"!" } {TEXT -1 1 "." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 274 10 "Problem 3:" }}{PARA 0 "" 0 "" {TEXT -1 24 "Consider the equation " }{XPPEDIT 18 0 "2*x^2*(1-x)*diff(y,`$`(x,2))-x*(1+x)*diff(y,x)+(1+x)*y = 0;" "6#/,(**\"\"#\"\"\"*$%\"xGF&F',&F'F'F)!\"\"F'-%%diffG6$%\"yG-% \"$G6$F)F&F'F'*(F)F',&F'F'F)F'F'-F-6$F/F)F'F+*&,&F'F'F)F'F'F/F'F'\"\"! " }{TEXT -1 3 ", " }{TEXT 322 1 "x" }{TEXT -1 5 " > 0." }}{PARA 0 "" 0 "" {TEXT -1 10 "(a) Is " }{TEXT 323 1 "x" }{TEXT -1 58 " = 0 a r egular point or a singular point of the equation?" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 82 "(b) Find the general s olution of the equation in terms of a series centered at " }{TEXT 324 1 "x" }{TEXT -1 5 " = 0." }}{PARA 0 "" 0 "" {TEXT -1 75 " _______ __________________________________________________________________" }} }{PARA 0 "" 0 "" {TEXT -1 129 " MSIP Grant #P120A80089-98: \"Three U rban Calculus Reform Programs: Adopting the Best,\" 1998-2001, Revi sed September 18, 2003" }}}{MARK "15 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }