{VERSION 5 0 "IBM INTEL NT" "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 }{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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 241 "This is the Maple Worksheet for analyzing the power series sol utions to Airy's Equation. Maple will also use another function known as the Gamma Function- You can look it up if you're curious- we had a side note about that function earlier." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "We' ll first define the differential equation twice to get a fundamental s et of solutions using Maple's built in functions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Execute these Maple comm ands starting from the beginning, and take note of what Maple is telli ng you" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "de:=diff(y(x),x$2)+x*y(x)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Y1:=dsolve(\{de,y(0)=1,D(y)(0)=0\},y(x));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "Y2:=dsolve(\{de,y(0)=0,D(y)(0)=1\},y(x));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y1:=rhs(Y1); y2:=rhs(Y2);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot([y1,y2],x=0..5,y=-1.. 2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 306 "Now we'll use Maple to get the Power series representations. \+ First we define the order of the series, then we solve the equation. \+ To plot the series, we have to convert the solutions (this is a Maple \+ thing). Finally, we compare the plots of Maple's built-in solutions t o our power series approximations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 104 "Here is the set of commands for order 7 \+ (the polynomial will go up to x^7- again, this is a Maple thing)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Order:=7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Y1PS:=dsolve(\{de,y(0)=1,D(y)(0)=0\},y(x),s eries);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "y1ps:=convert(rh s(Y1PS),polynom);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Y2PS:= dsolve(\{de,y(0)=0,D(y)(0)=1\},y(x),series);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "y2ps:=convert(rhs(Y2PS),polynom);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot([y1,y1ps],x=0..5,y=-1..2);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot([y2,y2ps],x=0..5,y=-1. .2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "These solutions are pretty good for x in (approximately) [0, 3/ 2]. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 " Homework: Find the lowest order for which the solutions are pretty go od on [0,4]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }