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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 258 25 "Separable First Order O
DE" }}{PARA 257 "" 0 "" {TEXT 256 47 "Date: Jan 16, 2002\nLast Revisio
n: Jan 16, 2002\n" }{TEXT 267 7 "Maple 6" }}{PARA 259 "" 0 "" {TEXT 
259 16 "Bent E. Petersen" }}{PARA 258 "" 0 "" {TEXT 260 17 "bent@alum.
mit.edu" }}{PARA 258 "" 0 "" {TEXT 261 22 "petersen@math.orst.edu" }}
{PARA 0 "" 0 "" {TEXT 262 0 "" }}{PARA 0 "" 0 "" {TEXT 263 15 "Course:
 Mth 256" }}{PARA 0 "" 0 "" {TEXT 264 17 "Term: Winter 2002" }}{PARA 
0 "" 0 "" {TEXT 265 11 "File name: " }{TEXT 257 26 "256w2002-separable
-ode.mws" }{TEXT 266 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 216 "This worksheet gives Maple's solutions to a few s
eparable ODEs. It demonstrates that you can use Maple to check your ho
mework solutions (though Maple has better uses). Maple is available in
 many campus computer labs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 92 "You should modify and play with this worksheet \+
to get anything useful out of it. Experiment!" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}
}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 
9 "Problem 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "ode01:=diff(y(t),t)=(y(t)+3)/(t+1);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%&ode01G/-%%diffG6$-%\"yG6#%\"tGF,*&,&F)\"\"\"
\"\"$F/F/,&F,F/F/F/!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"soln01:=dsolve(ode01,y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sol
n01G/-%\"yG6#%\"tG,&*&,&F)\"\"\"F-F-F-%$_C1GF-F-\"\"$!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Note Mapl
e uses  _C1  here for the arbitrary constant. If you prefer  c  you ca
n simply substitute." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "subs(_C1=c,soln01);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"tG,&*&,&F'\"\"\"F+F+F+%\"cGF+F+\"\"$!\"\"" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "%?(t) = y*c-3;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%#%?G6#%\"tG,&*&%\"yG\"\"\"%\"cGF+F+\"\"$!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
268 9 "Problem 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 50 "ode02:=diff(y(t),t)=(3*t+1)/(exp(y(t))+sin(y(
t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode02G/-%%diffG6$-%\"yG6#%
\"tGF,*&,&F,\"\"$\"\"\"F0F0,&-%$expG6#F)F0-%$sinGF4F0!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "dsolve(ode02,y(t)): subs(_C1=c,%);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*$)%\"tG\"\"#\"\"\"#\"\"$F(F'F)
-%$expG6#-%\"yG6#F'!\"\"-%$cosGF.F)%\"cGF)\"\"!" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 249 "Note here  %  refer
s to the previously evaluated expression (so, the output from dsolve).
 The output ftom dsolve is not visible because the command is terminat
ed by a colon. Thus you see only the result of substituting  c  for  _
C1  in the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 123 "If we wish to write the solution in a more conventiona
l form we can also substitute  y  for  y(t). Here's one way to do it:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "dsolve(ode02,y(t)): subs(_C1=c,y(t)=y,%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,,*$)%\"tG\"\"#\"\"\"#\"\"$F(F'F)-%$expG6#%
\"yG!\"\"-%$cosGF.F)%\"cGF)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 269 9 "Problem 3" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ode03:=diff
(y(x),x)=sqrt(y(x)+1)*sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&od
e03G/-%%diffG6$-%\"yG6#%\"xGF,*&-%%sqrtG6#,&F)\"\"\"F2F2F2-%$sinGF+F2
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "init03:=y(0)=2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'init03G/-%\"yG6#\"\"!\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "soln03:=dsolve(\{ode03,init0
3\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'soln03G/-%\"yG6#%\"xG
-%'RootOfG6#,*-%$cosGF(!\"\"*&\"\"#\"\"\"-%%sqrtG6#,&%#_ZGF3F3F3F3F0F3
F3*&F2F3-F56#\"\"$F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 97 "Note how we include the initial condition
s (we now have a set of equations as indicated by \{..\})." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 173 "The solution ret
urned by Maple means that  y(x)  is one, or more, of the roots of the \+
indicated expression in  _Z.  We can find these roots by using the  al
lvalues  command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 18 "allvalues(soln03);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"xG,,#\"\"*\"\"%\"\"\"*&#F,F+F,)-%$cosGF&\"
\"#F,F,*&#F,F2F,F0F,!\"\"*&F0F,-%%sqrtG6#\"\"$F,F5*$F7F,F," }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 9 "Problem 4
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "ode04:=diff(y(x),x)=(x-1)*(y(x)+1)^(1/3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&ode04G/-%%diffG6$-%\"yG6#%\"xGF,*&,&F,\"
\"\"F/!\"\"F/),&F)F/F/F/#F/\"\"$F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "dsolve(ode04,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/,**$)%\"xG\"\"#\"\"\"#F)F(F'!\"\"*&#\"\"$F(F)*$),&-%\"yG6#F'F)F)F)#
F(F.F)F)F+%$_C1GF)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 78 "Note Maple returned an implicit solution.
 We can ask for an explicit solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "soln04:=dsolve(ode04,y(
x),explicit=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'soln04G/-%\"y
G6#%\"xG-%'RootOfG6#,**$)F)\"\"#\"\"\"F1*&F0F1F)F1!\"\"*&\"\"$F1),&%#_
ZGF1F1F1#F0F5F1F3*&F0F1%$_C1GF1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "That expression may not appear hel
pful. Let's try the allvalues command" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "soln04a:=allvalues(soln
04);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(soln04aG6$/-%\"yG6#%\"xG,&!
\"\"\"\"\"*&#F-\"\"*F--%%sqrtG6#,6*$)F*\"\"&F-!#=*(\"#=F-)F*\"\"%F-%$_
C1GF-F-*&\"\"$F-)F*\"\"'F-F-*(\"#OF-)F*\"\"#F-)F=FEF-F-*&\"#CF-)F*F?F-
F,*&FCF-F;F-F-*(\"#sF-FIF-F=F-F,*&FHF-)F=F?F-F-*(FLF-FDF-F=F-F-*(FLF-F
*F-FFF-F,F-F-/F',&F,F-*&#F-F0F-*$F1F-F-F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Note we have two expressi
ons here. Let's substitute  _C1=c  in each to make them easier to read
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "subs(_C1=c,soln04a[1]); subs(_C1=c,soln04a[2]);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&!\"\"\"\"\"*&#F*\"\"*F
*-%%sqrtG6#,6*$)F'\"\"&F*!#=*(\"#=F*)F'\"\"%F*%\"cGF*F**&\"\"$F*)F'\"
\"'F*F**(\"#OF*)F'\"\"#F*)F:FBF*F**&\"#CF*)F'F<F*F)*&F@F*F8F*F**(\"#sF
*FFF*F:F*F)*&FEF*)F:F<F*F**(FIF*FAF*F:F*F**(FIF*F'F*FCF*F)F*F*" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&!\"\"\"\"\"*&#F*\"\"*F
**$-%%sqrtG6#,6*$)F'\"\"&F*!#=*(\"#=F*)F'\"\"%F*%\"cGF*F**&\"\"$F*)F'
\"\"'F*F**(\"#OF*)F'\"\"#F*)F;FCF*F**&\"#CF*)F'F=F*F)*&FAF*F9F*F**(\"#
sF*FGF*F;F*F)*&FFF*)F;F=F*F**(FJF*FBF*F;F*F**(FJF*F'F*FDF*F)F*F*F)" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "P
erhaps you prefer the original implicit solution - cleaned up a bit:" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 39 "subs(_C1=c,y(x)=y, dsolve(ode04,y(x)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,**$)%\"xG\"\"#\"\"\"#F)F(F'!\"\"*&#\"\"$F(F)*$),&%\"y
GF)F)F)#F(F.F)F)F+%\"cGF)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 271 9 "Problem 5" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ode05:=diff(s(t),
t)=exp(s(t)+t)/(t+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode05G/-%%
diffG6$-%\"sG6#%\"tGF,*&-%$expG6#,&F)\"\"\"F,F2F2,&F,F2F2F2!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "init05:=s(1)=1;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%'init05G/-%\"sG6#\"\"\"F)" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 36 "soln05:=dsolve(\{ode05,init05\},s(t));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'soln05G/-%\"sG6#%\"tG-%#lnG6#,$*&\"
\"\"F/,&*&-%$expG6#!\"\"F/-%#EiG6$F/,&F)F5F/F5F/F5*&,&*(-F36#F/F/-F76$
F/!\"#F/F2F/F/F/F5F/F=F5F/F5F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 188 "The function Ei() is the exponential
 integral (see Maple help). The constant  I  is the square root of  -1
. This solution is difficult to visualize. We can plot a bit of it to \+
get insight:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "expr05:=rhs(soln05);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'expr05G-%#lnG6#,$*&\"\"\"F*,&*&-%$expG6#!\"\"F*-%#Ei
G6$F*,&%\"tGF0F*F0F*F0*&,&*(-F.6#F*F*-F26$F*!\"#F*F-F*F*F*F0F*F9F0F*F0
F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(expr05,t=0..5/4)
;" }}{PARA 13 "" 1 "" {GLPLOT2D 389 292 292 {PLOTDATA 2 "6%-%'CURVESG6
$7fn7$$\"\"!F)$!+%\\`'4S!#57$$\"+rUkCF!#6$!+.N[DQF,7$$\"+$oU`4&F0$!+``
KiOF,7$$\"+\"*4WhxF0$!+g_OvMF,7$$\"+$>@X/\"F,$!+IZ>$G$F,7$$\"+PFi68F,$
!+\\0q(3$F,7$$\"+g+Ef:F,$!+j*=C!HF,7$$\"+kTn:=F,$!+Q(4iq#F,7$$\"+)*z&3
3#F,$!+\\iJ)\\#F,7$$\"+$Q\">XBF,$!+])>dG#F,7$$\"+h24<EF,$!+$=j51#F,7$$
\"+\"))zl&GF,$!+^W$y&=F,7$$\"+2z=EJF,$!+!)zkA;F,7$$\"+pH!pR$F,$!+'e:#z
8F,7$$\"+%4(ydOF,$!+ZgBP6F,7$$\"+t[p%*QF,$!+Xb&y5*F07$$\"+O6SwTF,$!+\\
e;EjF07$$\"+#HV]T%F,$!+k!H)))QF07$$\"+_3k#p%F,$!+JC)H`*!#77$$\"+I<OQ\\
F,$\"+;DuT<F07$$\"+xk&z?&F,$\"+1o76[F07$$\"+IWnkaF,$\"+u.%=&yF07$$\"+$
oKDt&F,$\"+*\\Fd6\"F,7$$\"+v,^yfF,$\"++s>K9F,7$$\"+J,$QC'F,$\"+fTi)y\"
F,7$$\"+)*HU>lF,$\"+4.\"p<#F,7$$\"+*RF$fnF,$\"+%H(GJDF,7$$\"+D2V=qF,$
\"+#R#yKHF,7$$\"+8)4hG(F,$\"+D50qLF,7$$\"+z:)za(F,$\"+GVIAQF,7$$\"+yfN
,yF,$\"+;vZ&G%F,7$$\"+[]o#3)F,$\"+]IwK[F,7$$\"+I<ZN$)F,$\"+a8%yN&F,7$$
\"+qaP0')F,$\"+M4*z&fF,7$$\"+u$[*\\))F,$\"+`'*)=a'F,7$$\"+d6L<\"*F,$\"
+&[n.B(F,7$$\"+>v\"*o$*F,$\"+N#*[LzF,7$$\"+fx*=j*F,$\"+&o'yN()F,7$$\"+
7!4!*))*F,$\"+=T@*f*F,7$$\"+n!=e,\"!\"*$\"+*Gk/1\"Fjw7$$\"+'\\U</\"Fjw
$\"+2fUp6Fjw7$$\"+pRDo5Fjw$\"+%pxkH\"Fjw7$$\"+9fa%4\"Fjw$\"+s+*GW\"Fjw
7$$\"+\"[0(=6Fjw$\"+M*f;g\"Fjw7$$\"+s]RY6Fjw$\"+@_rC=Fjw7$$\"+r4;r6Fjw
$\"+Rfu\"3#Fjw7$$\"+yPO%=\"Fjw$\"+^Fa`AFjw7$$\"+&emv>\"Fjw$\"+,EwiCFjw
7$$\"+cL?57Fjw$\"+P\"onr#Fjw7$$\"+G,%GA\"Fjw$\"+29$41$Fjw7$$\"+'4I'H7F
jw$\"+EI&=J$Fjw7$$\"+k+UO7Fjw$\"+'p;\"\\OFjw7$$\"+[]\")R7Fjw$\"+`.mtQF
jw7$$\"+K+@V7Fjw$\"+<MJkTFjw7$$\"+Cv!\\C\"Fjw$\"+3YI\\VFjw7$$\"+;]gY7F
jw$\"+ehqwXFjw7$$\"+3DI[7Fjw$\"+`'[=([Fjw7$$\"++++]7Fjw$\"+%)o*GH&Fjw-
%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"t6\"Q!6\"-%%VIEWG6$;
F(Fc]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 
0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 78 "Another way to obtain information about solutions is t
o expand in series. Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "dsolve(\{ode05,init05\},s(t),type=s
eries);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"sG6#%\"tG+1,&F'\"\"\"F
*!\"\"F*\"\"!,$-%$expG6#F*#F*\"\"#F*,&*$)F.F2F*#F*\"\")*&F6F*F.F*F*F2,
(*$)F.\"\"$F*#F*\"#C*&#F*\"#;F*F5F*F**&F=F*F.F*F*F<,**$)F.\"\"%F*#F*\"
#k*&#F*\"#KF*F;F*F**&#\"#6\"$%QF*F5F*F**&#F*\"$#>F*F.F*F*FF,,*$)F.\"\"
&F*#F*\"$g\"*&FGF*FEF*F**&#\"\"(FOF*F;F*F**&#F*\"$G\"F*F5F*F**&#F*\"$!
[F*F.F*F*FV-%\"OGF0\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 92 "We can get a more readable expression by \+
evaluating the constants as floating point numbers." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(%,5
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"sG6#%\"tG+1,&F'\"\"\"$F*\"
\"!!\"\"$F*F,F,$\"&#f8!\"%F*$\"&ME\"F1\"\"#$\"&?T\"F1\"\"$$\"&nq\"F1\"
\"%$\"&-@#F1\"\"&-%\"OG6#F*\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "If you solve this problem by hand \+
you probably will obtain the solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "hand05:=s(t)=-log(exp(-
1)-Int(exp(r)/(r+1),r=1..t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ha
nd05G/-%\"sG6#%\"tG,$-%#lnG6#,&-%$expG6#!\"\"\"\"\"-%$IntG6$*&-F06#%\"
rGF3,&F:F3F3F3F2/F:;F3F)F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 36 "Maple can evaluate the integral here" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "hand05:=s(t)=-log(exp(-1)-int(exp(r)/(r+1),r=1..t));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'hand05G/-%\"sG6#%\"tG,$-%#lnG6#,(-%$expG6
#!\"\"\"\"\"*&F/F3-%#EiG6$F3,&F)F2F3F2F3F3*&-F66$F3!\"#F3F/F3F2F2" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "T
his solution is the same as obtained above as we can see by comparing \+
the exponentials of the right hand sides of the solution equations:" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 47 "exp(rhs(soln05))-exp(rhs(hand05)): simplify(%);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 272 9 "Problem 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ode06:=diff(u(t),t)=(se
c(u(t)))^2/(1+t^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode06G/-%%di
ffG6$-%\"uG6#%\"tGF,*&*$)-%$secG6#F)\"\"#\"\"\"F4,&F4F4*$)F,F3F4F4!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "soln06:=dsolve(ode06,
u(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'soln06G/-%\"uG6#%\"tG,$-%
'RootOfG6#,*%#_ZG\"\"\"*&\"\"%F0-%'arctanGF(F0!\"\"-%$sinG6#F/F0*&F2F0
%$_C1GF0F5#F0\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 52 "The  allvalues command will not help here (try it
). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "T
he meaning of the expression  soln06  is that  2u  is a root of the ex
pression in  _Z  in the argument of the  RootOf  command.  That is" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "2*u-4*arctan(t)+sin(2*u)-4*c=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/,*%\"uG\"\"#*&\"\"%\"\"\"-%'arctanG6#%\"tGF)!\"\"-%$sinG6#,$F%F&F)
*&F(F)%\"cGF)F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}}{MARK "0 13 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 
1 2 33 1 1 }