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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 19 "ENGI 3424 Chapter 1" }
{TEXT -1 51 " - Demonstration of Maple for the solution of some " }}
{PARA 0 "" 0 "" {TEXT -1 78 "first order linear ordinary differential \+
equations with constant coefficients," }}{PARA 0 "" 0 "" {TEXT -1 38 "
using examples from the lecture notes." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
13 "Example 1.1.2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ode :=
 diff(x(t),t) = k*x(t)*(1-x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$odeG/-%%diffG6$-%\"xG6#%\"tGF,*(%\"kG\"\"\"F)F/,&F/F/F)!\"\"F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(ode, x(t));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG*&\"\"\"F),&F)F)*&-%$expG6#,$*
&%\"kGF)F'F)!\"\"F)%$_C1GF)F)F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
13 "Example 1.1.3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ode :=
 m*diff(v(t),t) = m*g - b*v(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$odeG/*&%\"mG\"\"\"-%%diffG6$-%\"vG6#%\"tGF/F(,&*&F'F(%\"gGF(F(*&%\"b
GF()F,\"\"#F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ics :
= v(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$icsG/-%\"vG6#\"\"!F)" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "dsolve(\{ode,ics\}, v(t))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"vG6#%\"tG*(-%%tanhG6#*(F'\"
\"\"*(%\"mGF-%\"gGF-%\"bGF-#F-\"\"#F/!\"\"F-F.F2F1F4" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 13 "Note:   tanh " }{TEXT 257 1 "x" }{TEXT -1 8 " =
 sinh " }{TEXT 258 1 "x" }{TEXT -1 8 " / cosh " }{TEXT 259 1 "x" }
{TEXT -1 6 " = (1 " }{TEXT 261 1 "-" }{TEXT -1 5 " exp(" }{TEXT 262 1 
"-" }{TEXT -1 1 "2" }{TEXT 260 1 "x" }{TEXT -1 14 ")) / (1 + exp(" }
{TEXT 265 1 "-" }{TEXT -1 1 "2" }{TEXT 263 1 "x" }{TEXT -1 4 ")) ," }}
{PARA 0 "" 0 "" {TEXT -1 78 "so that the answer above is consistent wi
th the solution in the lecture notes." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 56 "Example 1.2.4 (water filling a leaking cylindrical tank)
" }}{PARA 0 "" 0 "" {TEXT -1 58 "First, the roles of height and time a
re reversed, so that " }{TEXT 266 1 "h" }{TEXT -1 28 " is the independ
ent variable" }}{PARA 0 "" 0 "" {TEXT -1 70 "and an explicit solution \+
is found for time as a function of height.   " }}{PARA 0 "" 0 "" 
{TEXT -1 40 "The inverse hyperbolic tangent, arctanh(" }{TEXT 267 1 "x
" }{TEXT -1 24 "), is also (1/2) ln ((1+" }{TEXT 270 1 "x" }{TEXT -1 
4 ")/(1" }{TEXT 271 1 "-" }{TEXT 269 1 "x" }{TEXT -1 6 ")).   " }}
{PARA 0 "" 0 "" {TEXT -1 88 "One can show that the solution below is e
quivalent to the solution in the lecture notes." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 44 "ode := A / diff(t(h),h) = Q - a*sqrt(2*g*h);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/*&%\"AG\"\"\"-%%diffG6$-%\"t
G6#%\"hGF/!\"\",&%\"QGF(*(%\"aGF(\"\"##F(F5*&%\"gGF(F/F(F6F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ics := t(h0) = 0;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$icsG/-%\"tG6#%#h0G\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "dsolve(\{ode,ics\}, t(h));" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#/-%\"tG6#%\"hG,.*&#\"\"\"\"\"#F+*,%\"AGF+%\"QGF+
-%#lnG6#,&*$)F/F,F+!\"\"**F,F+)%\"aGF,F+%\"gGF+F'F+F+F+F9!\"#F:F6F+F6*
,F.F+F9F6F,#F+F,F:F6*&F:F+F'F+F=F6*,F.F+F9F;F:F6F/F+-%(arctanhG6#**F9F
+F>F=F,F=F/F6F+F+*&F=F+*,F.F+F/F+-F16#,&F4F6**F,F+F8F+F:F+%#h0GF+F+F+F
9F;F:F6F+F+*,F.F+F9F6F,F=F:F6*&F:F+FJF+F=F+*,F.F+F9F;F:F6F/F+-FA6#**F9
F+FLF=F,F=F/F6F+F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "However, Ma
ple struggles to find an explicit solution for height as a function of
 time:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "ode := A * diff(h
(t),t) = Q - a*sqrt(2*g*h(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$o
deG/*&%\"AG\"\"\"-%%diffG6$-%\"hG6#%\"tGF/F(,&%\"QGF(*(%\"aGF(\"\"##F(
F4*&%\"gGF(F,F(F5!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "i
cs := h(0) = h0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$icsG/-%\"hG6#\"
\"!%#h0G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "dsolve(\{ode,ic
s\}, h(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"tG-%'RootOf
G6#,0**\"\"#\"\"\"F'F.)%\"aGF-F.%\"gGF.F.*(%\"AGF.%\"QGF.-%#lnG6#,&*$)
F4F-F.!\"\"**F-F.F/F.F1F.%#_ZGF.F.F.F.*,F-F.F3F.F-#F.F-*&F1F.F=F.F?F0F
.F.**F-F.F3F.F4F.-%(arctanhG6#**F0F.F@F?F-F?F4F;F.F;*(F3F.F4F.-F66#,&F
9F;**F-F.F/F.F1F.%#h0GF.F.F.F;*,F-F.F3F.F-F?*&F1F.FKF.F?F0F.F;**F-F.F3
F.F4F.-FC6#**F0F.FMF?F-F?F4F;F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
13 "Example 1.3.1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ode := diff(y(x),x) + \+
2*y(x) = 6*exp(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,&-%%dif
fG6$-%\"yG6#%\"xGF-\"\"\"*&\"\"#F.F*F.F.,$*&\"\"'F.-%$expGF,F.F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(ode, y(x));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"#\"\"\"-%$expGF&F+F+*&
-F-6#,$*&F*F+F'F+!\"\"F+%$_C1GF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 13 "Example 1.3.3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ode
 := diff(y(x),x) - y(x) = sinh(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$odeG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"F*!\"\"-%%sinhGF," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(ode, y(x));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&-%$expGF&\"\"\",(*&\"\"#!\"\"F'F
,F,*&#F,\"\"%F,-%%sinhG6#,$*&F/F,F'F,F,F,F0*&#F,F3F,-%%coshGF6F,F,F,F,
*&F*F,%$_C1GF,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Example 1.3.4
 - (linear only for " }{TEXT 272 1 "x" }{TEXT -1 18 " as a function of
 " }{TEXT 273 1 "y" }{TEXT -1 1 ")" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "ode := (1 - 2*x*exp(2*y(x)))*diff(y(x),x) = exp(2*y(x
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/*&,&\"\"\"F(*(\"\"#F(%
\"xGF(-%$expG6#,$*&F*F(-%\"yG6#F+F(F(F(!\"\"F(-%%diffG6$F1F+F(F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(ode, y(x));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"#F+-%)LambertWG
6#,$*(F,F+F'F+-%$expG6#%$_C1G!\"#!\"\"F+F7F5F7" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 53 "ode := (1 - 2*x(y)*exp(2*y))/diff(x(y),y) = ex
p(2*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/*&,&\"\"\"F(*(\"\"#
F(-%\"xG6#%\"yGF(-%$expG6#,$*&F*F(F.F(F(F(!\"\"F(-%%diffG6$F+F.F4F/" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(ode, x(y));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"yG*&,&F'\"\"\"%$_C1GF*F*-%
$expG6#,$*&\"\"#F*F'F*!\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "
Example 1.4.2 - reduction of order" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "ode := diff(y(x),x,x) = (diff(y(x),x))^2;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$odeG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"
\"#*$)-F'6$F)F,F0\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "
dsolve(ode, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$
-%#lnG6#,&*&%$_C1G\"\"\"F'F/!\"\"%$_C2GF0F0" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 34 "Example 1.4.3 - reduction of order" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 44 "ode := diff(y(x),x,x) = 2*y(x)*diff(y(x),x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/-%%diffG6$-%\"yG6#%\"xG-%
\"$G6$F,\"\"#,$*(F0\"\"\"F)F3-F'6$F)F,F3F3" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "dsolve(ode, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%\"yG6#%\"xG*&-%$tanG6#*&,&F'\"\"\"%$_C2GF.F.%$_C1G!\"\"F.F0F1" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Maple misses the singular soluti
on, when  " }{TEXT 274 3 "_C1" }{TEXT -1 5 " = 0." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 1" 51 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }