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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Problem SetC #8 by " }}
{PARA 0 "" 0 "" {TEXT -1 32 "Gabriel Johnson and Chris Deyton" }}
{PARA 0 "" 0 "" {TEXT -1 114 "We are given the DE : y'=e^(-x^2) and we
 are told that the DE cannot be solved inn terms of elementary functio
ns. " }}{PARA 0 "" 0 "" {TEXT -1 5 "A.)  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We used dsolve to solve th
e DE." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "dsolve(diff(y(x),x
)=exp(-x^2),y(x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&-%%sqrtG6#%#PiG\"\"\"-%$erfGF&
\"\"\"#F1\"\"#%$_C1GF1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "B.)" }}
{PARA 0 "" 0 "" {TEXT -1 178 "When we solved the DE we encountered the
 built in function erf.  We are asked to show that when we take the de
rivative of erf(x) with respect to x we get (2/sqrt(Pi))*e^(-x^2).. " 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(erf(x),x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*&-%$expG6#,$*$)%\"xG\"\"#\"\"\"!\"\"F-*$-
%%sqrtG6#%#PiGF-!\"\"F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "C.)" }}
{PARA 0 "" 0 "" {TEXT -1 325 "We are asked to illustrate the numerical
 capabilities of Maple to show that even though we don't have the elem
entary formulas for this function we \"know\" this function as well as
 we \"know\" other elementary functions.  In order to do this we evalu
ated erf(x) at x=0,1,10.5 and then plotted erf(x) where x ranges from \+
-10 to10." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "evalf(erf(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(erf(1));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Hz+F%)!#5" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "evalf(erf(10.5));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(
plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(erf(x),x=-1
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$\"1-+]P?Wl&*F0F`[l7$$\"#5F*F`[l-%'COLOURG6&%$RGBG$F\\]lF,F*F*-%+AXESL
ABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(F[]l%(DEFAULTG" 2 156 138 138 2 0 1 0 
2 9 1 4 2 1 45 45 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 
0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "D.)" }}{PARA 0 "" 0 "" 
{TEXT -1 134 "We asked to compute the limit of erf(x) as x-> infinity \+
and we are asked to find the integral of e^(-t^2) from t=-infinity...i
nfinity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "limit(erf(x),x=infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "As you c
an see this agrees with the graph above of erf(x)." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "int(exp(-t^
2),t=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqr
tG6#%#PiG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "The above com
mand is the same as if we had solved the original DE by integrating fr
om -infinity to infinity." }}{PARA 0 "" 0 "" {TEXT -1 3 "E.)" }}{PARA 
0 "" 0 "" {TEXT -1 189 "We were asked to solve the initial value probl
em y'=1-2xy, y(0)=0 using the Maple command dsolve.  We unapplied at t
he same time as dsolving because we are going to use this solution lat
er." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "sol:=unapply(rhs(dsolve(\{diff(y(x),x)=1-2*x*y(x),y(0
)=0\},y(x))),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solGR6#%\"xG6\"
6$%)operatorG%&arrowGF(,$**%\"IG\"\"\"-%$expG6#,$*$)9$\"\"#F/!\"\"\"\"
\"-%%sqrtG6#%#PiGF/-%$erfG6#*&F.F9F6F9F9#F8F7F(F(F(" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 135 "We are asked what happens to the solution for \+
large x.  If we look at the function y(x) when x is extremely large y(
x) approaches zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 81 "We are also asked to find the value of x at whi
ch the solution is at its maximum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "deriv:=unapply(diff(sol(x),
x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&derivGR6#%\"xG6\"6$%)oper
atorG%&arrowGF(,&*,%\"IG\"\"\"9$F/-%$expG6#,$*$)F0\"\"#F/!\"\"\"\"\"-%
%sqrtG6#%#PiGF/-%$erfG6#*&F.F9F0F9F9F9*&F1F/-F26#F5F9F9F(F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "We unapplied the derivative so th
at we could get it as a function of x.  We are also asked to find the \+
value where the solution is at its maximum." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(sol(x),x=0..
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}{EXCHG {PARA 0 "" 0 "" {TEXT -1 328 "The graph above wasn't required,
 but it is a good illustration of the sol.  This is the graph of the s
olution found when we dsolved the DE we were given.  You can see from \+
the graph that the x value where the solution takes its maximum is aro
und  0.91.  We used fsolve to get the exact value for x where the solu
tion is a max.  " }}{PARA 0 "" 0 "" {TEXT -1 83 "We tried using the so
lve command instead of fsolve but it didn't give us an answer." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fsolve(deriv(x)=0,x=.5..2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+I()QT#*!#5" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "evalf(sol(.9241388730));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+ZAW5a!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "
The above solution is the max value for the " }{TEXT 256 8 "solution" 
}{TEXT -1 47 ".  This answer also jives with the graph above." }}
{PARA 0 "" 0 "" {TEXT -1 3 "F.)" }}{PARA 0 "" 0 "" {TEXT -1 89 "We are
 asked to compute the partial derivative of f with respect to y for f(
x,y)=1-2xy.  " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "diff(1-2*x*y,y);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$%\"xG!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "This is the \+
same as what we got by hand." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 69 "iniset:=\{[y(0)=.5],[y(0)=1],[y(0)=1.5],[y(0)=2],[y(0)=2.5],[y(0
)=3]\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "DEplot((diff(y(x),x)=1-2*x*y(x),y
(x)),x=-3..3,iniset,linecolor=green);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
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1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1027 "To ma
ke things more convenient we plotted from -3...3 instead of [-3,0] and
 [0,3] seperately as the book wanted.  In this part of the problem we \+
are asked to discuss the stability.  As far as stability from -3..0 th
e graph appears to show  that the solutions aren't very stable because
 the field lines are turning away from the plot.  However of the right
 side as the graph approaches the x axis after x=1.5 the graph appears
 to be forming some stability because the field lines are merging towa
rds the solution curves.  We tried one argument that showed us that th
eir weren't any horizontal line solutions.  We have the DE y'=1-2xy an
d the solutions will be stable if y is a contant.  If y is constant th
en 1-2xy=0 because the derivative of a constant is 0.  So we have 1-2x
(constant)=0 and this isn't possible for all values of x.  Therefore, \+
we don't have any horizontal line solutions.  I wasn't really sure wha
t would happen with the stability so I didn't have a conclusion about \+
the stability until I looked at the graph." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT -1 133 "We would really like to know what the equ
ation of the lines are when the all merge together, but we didn't have
 much success at this." }}}}{MARK "0 1 0" 13 }{VIEWOPTS 1 1 0 3 2 
1804 }