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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 25 "Problem Set D, Problem 1.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 50 "We consider the second order differential
 equation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "airy := diff(y(x), x$2
) = x*y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%airyG/-%%diffG6$-F'6
$-%\"yG6#%\"xGF.F.*&F.\"\"\"F+F0" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 
3 "(a)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 266 "When x is close to 0, \+
we want to compare the solution of Airy's equation with the solution o
f the IVP  y''(x) = 0,  y'(0) = 1,  y(0) = 0. The solution of this IVP
 is y = x. Here is a plot of the numerical solution of Airy's equation
 together with the function y = x." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "with(plots): with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 31 "fac1plot := plot(x, x = -2..2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 106 "airy1plot := DEplot(airy, y(x), x = -2..2, \{[y(0) =
 0, D(y)(0) = 1]\}, \nmethod = rkf45, linecolor = black):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "display(\{fac1plot, airy1plot\});" 
}}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7S7$$!\"#\"\"!F(7$$!1LLL$Q6
G\">!#:F,7$$!1nm;M!\\p$=F.F07$$!1LLL))Qj^<F.F37$$!1LLL=Kvl;F.F67$$!1nm
;C2G!e\"F.F97$$!1LL$3yO5]\"F.F<7$$!1++]nU)*=9F.F?7$$!1LL$3WDTL\"F.FB7$
$!1++]d(Q&\\7F.FE7$$!1nmmc4`i6F.FH7$$!1LLLQW*e3\"F.FK7$$!1,+++()>'***!
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j^NmFPFen7$$!1ommm9'=(eFPFhn7$$!1,++v#\\N)\\FPF[o7$$!1pmmmCC(>%FPF^o7$
$!1*****\\FRXL$FPFao7$$!1+++D=/8DFPFdo7$$!1mmm;a*el\"FPFgo7$$!1pmm;Wn(
o)!#<Fjo7$$!1qLLL$eV(>!#=F^p7$$\"1Mmm;f`@')F\\pFbp7$$\"1)****\\nZ)H;FP
Fep7$$\"1lmm;$y*eCFPFhp7$$\"1*******R^bJ$FPF[q7$$\"1'*****\\5a`TFPF^q7
$$\"1(****\\7RV'\\FPFaq7$$\"1'*****\\@fkeFPFdq7$$\"1JLLL&4Nn'FPFgq7$$
\"1*******\\,s`(FPFjq7$$\"1lmm\"zM)>$)FPF]r7$$\"1*******pfa<*FPF`r7$$
\"1HLLeg`!)**FPFcr7$$\"1++]#G2A3\"F.Ffr7$$\"1LLL$)G[k6F.Fir7$$\"1++]7y
h]7F.F\\s7$$\"1nmm')fdL8F.F_s7$$\"1nmm,FT=9F.Fbs7$$\"1LL$e#pa-:F.Fes7$
$\"1+++Sv&)z:F.Fhs7$$\"1LLLGUYo;F.F[t7$$\"1nmm1^rZ<F.F^t7$$\"1++]sI@K=
F.Fat7$$\"1++]2%)38>F.Fdt7$$\"\"#F*Fgt-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*
-F$6&777$$!+++++?!\"*$!19$>Ch*z\"**)FP7$$!+++++=Ffu$!1NKtx!)3R5F.7$$!+
++++;Ffu$!19$[aGmZ5\"F.7$$!+++++9Ffu$!1a*H\"fa:+6F.7$$!+++++7Ffu$!1w$z
c\\uT.\"F.7$$!+++++5Ffu$!1MpX!)*)G'=*FP7$$!+++++!)!#5$!1^PmlT!Gm(FP7$$
!+++++gFew$!15$34:aD*eFP7$$!+++++SFew$!11\">=<*pyRFP7$$FeuFew$!1on$H#p
m)*>FP7$F*F*7$$\"+++++?Few$\"1Mfk*eL8+#FP7$$\"+++++SFew$\"1WKb%)eO@SFP
7$$\"+++++gFew$\"1v^\"f\"ob3hFP7$$\"+++++!)Few$\"1:I@w\"=bM)FP7$$\"+++
++5Ffu$\"1oH1e'R`3\"F.7$$\"+++++7Ffu$\"1)R&3Bv/!Q\"F.7$$\"+++++9Ffu$\"
18rgexpT<F.7$$\"+++++;Ffu$\"1rCqlI)=?#F.7$$\"+++++=Ffu$\"1iKQ$>XW!GF.7
$$FixFfu$\"1aaxlP26OF.-%&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"$-Fjt6&F\\u
F*F*F*-%+AXESLABELSG6$%\"xG%!G-%%VIEWG6$;$!+++++AFfu$\"+++++AFfu;$!+)G
e0M\"Ffu$\"+pd'o%QFfu" 2 241 241 241 2 0 1 0 2 9 0 4 2 1.000000 
45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 
0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "The facsimile solutio
n agrees well with the actual solution in a neighborhood of 0." }}}
{EXCHG {PARA 4 "" 0 "" {TEXT -1 3 "(b)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 15 "For x close to " }{XPPEDIT 18 0 "-16 = -(4^2)" "/,$\"#;!
\"\",$*$\"\"%\"\"#F%" }{TEXT -1 78 ", we want to compare the solution \+
of Airy's equation to the facsimile solution" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "fac2 := (c1, c2) -> c1*sin(4*x + c2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%fac2G:6$%#c1G%#c2G6\"6$%)operatorG%&arrowGF)*&9
$\"\"\"-%$sinG6#,&%\"xG\"\"%9%F/F/F)F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 57 "First we'll plot a numerical solution of Airy's equation.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "airy2plot := DEplot(airy, y(x)
, -18..-14, \{[y(0) = 0, D(y)(0) = 1]\}, \nmethod = rkf45, stepsize = \+
0.1, linecolor = black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 
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{TEXT -1 131 "This certainly looks like a sine wave. Let's see how wel
l it matches up with an appropriate sine wave. We have to choose const
ants " }{XPPEDIT 18 0 "c[1]" "&%\"cG6#\"\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "c[2]" "&%\"cG6#\"\"#" }{TEXT -1 64 " in the facsimile s
olution to make it match up. Note first that " }{XPPEDIT 18 0 "c[1]" "
&%\"cG6#\"\"\"" }{TEXT -1 138 " is the amplitude of the facsimile solu
tion, and we can see from the graph that the amplitude of the solution
 of Airy's equation is about " }{XPPEDIT 18 0 ".61" "$\"#h!\"#" }
{TEXT -1 32 " on this interval. The constant " }{XPPEDIT 18 0 "c[2]" "
&%\"cG6#\"\"#" }{TEXT -1 148 " determines the phase shift, and can be \+
read off from the zeros of the solution. In particular, the solution o
f Airy's equation has a zero at about " }{XPPEDIT 18 0 "-16.3" ",$$\"$
j\"!\"\"!\"\"" }{TEXT -1 22 ", so we should choose " }{XPPEDIT 18 0 "c
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$\"1Uf83_z)4'F-7$$!1+v=TH;K9F^x$\"1OC0i^?*3'F-7$$!1M$3_Is6V\"F^x$\"1^z
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$!1m;/\"f.5U\"F^x$\"1&z8Ku!)fL&F-7$$!1++v#pynT\"F^x$\"1:f&\\S0Gw%F-7$$
!1+++E\\t79F^x$\"1bj&z*p%o3%F-7$$!1++Df6p39F^x$\"1!f9\\>'>/LF-7$$!1+]i
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\\bn;$!+iIx#)p!#5$\"+aFC+qFbbn" 2 241 241 241 2 0 1 0 2 9 0 4 2 
1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 
0 0 0 -18324 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "The pl
ots match up well. Why did we have to choose the values of " }
{XPPEDIT 18 0 "c[1]" "&%\"cG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "c[2]" "&%\"cG6#\"\"#" }{TEXT -1 104 " by hand? Our analysis sugg
ests that the facsimile solution should approximate the actual solutio
n near " }{XPPEDIT 18 0 "x = -K^2" "/%\"xG,$*$%\"KG\"\"#!\"\"" }{TEXT 
-1 77 ". But the initial condition that we used for Airy's equation is
 at the point " }{XPPEDIT 18 0 "x = 0" "/%\"xG\"\"!" }{TEXT -1 308 ", \+
where the facsimile solution is not a good approximation. Thus, the un
determined constants in the facsimile solution are unrelated to the in
itial conditions in Airy's equation, and we needed to choose them by h
and to get a good match. Nevertheless, the frequency of the facsimile \+
solution is determined by " }{XPPEDIT 18 0 "K" "I\"KG6\"" }{TEXT -1 
159 ", and is independent of the undetermined constants. Thus, we can \+
at least conclude that the frequency of the solutions of Airy's equati
on in a neighborhood of " }{XPPEDIT 18 0 "x = -K^2" "/%\"xG,$*$%\"KG\"
\"#!\"\"" }{TEXT -1 32 " will be almost proportional to " }{XPPEDIT 
18 0 "K" "I\"KG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 4 "" 0 "" {TEXT 
-1 3 "(c)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fac3 := (c1, c
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$%#c1G%#c2G6\"6$%)operatorG%&arrowGF)*&9$\"\"\"-%%sinhG6#,&%\"xG\"\"%9
%F/F/F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "airy3plot := \+
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45, linecolor = black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "
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{PARA 0 "" 0 "" {TEXT -1 99 "We'd like to compare this with the graph \+
of the hyperbolic sine function. Again, we have to choose " }{XPPEDIT 
18 0 "c[1]" "&%\"cG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c[2]
" "&%\"cG6#\"\"#" }{TEXT -1 48 " by hand. We'll start with an arbitrar
y choice: " }{XPPEDIT 18 0 "c[1] = 1" "/&%\"cG6#\"\"\"\"\"\"" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "c[2] = 0" "/&%\"cG6#\"\"#\"\"!" }{TEXT -1 1 
"." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(fac3(1, 0), x = 14..18);
" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7fn7$$\"#9\"\"!$\"1)\\1![
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" {TEXT -1 96 "The graphs are remarkably similar. Note, however, that \+
the values in the second graph are about " }{XPPEDIT 18 0 "10^9" "*$\"
#5\"\"*" }{TEXT -1 67 " greater than in the first, which means that we
 should have chosen " }{XPPEDIT 18 0 "c[1]" "&%\"cG6#\"\"\"" }{TEXT 
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