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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 258 22 "Linear First Order ODE
" }}{PARA 257 "" 0 "" {TEXT 256 47 "Date: Jan 20, 2002\nLast Revision:
 Jan 20, 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 25 "256w2002-linear-1-
ode.mws" }{TEXT 266 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 121 "This worksheet gives Maple's solutions to a few lin
ear first order ODEs. Maple is available in many campus computer labs.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "You s
hould modify and play with this worksheet to get anything useful out o
f 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 260 "" 0 "" {TEXT -1 9 "Problem 1" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ode01:=diff
(y(x),x)-4*x*y(x)=2*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode01G/,&
-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*(\"\"%F.F-F.F*F.!\"\",$F-\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dsolve(ode01,y(x)): subs(y(x
)=y,_C1=C,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&#!\"\"\"\"#\"
\"\"*&-%$expG6#,$*$)%\"xGF(F)F(F)%\"CGF)F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 2" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ode02:=d
iff(u(t),t)=(u(t)/t)+t^5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode02G
/-%%diffG6$-%\"uG6#%\"tGF,,&*&F)\"\"\"F,!\"\"F/*$)F,\"\"&F/F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dsolve(ode02,u(t)): subs(_C1
=C,u(t)=u,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"uG*&,&*$)%\"tG\"
\"&\"\"\"#F+F*%\"CGF+F+F)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ode03:=diff(x(t),t)+sec
(t)*x(t)=tan(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode03G/,&-%%dif
fG6$-%\"xG6#%\"tGF-\"\"\"*&-%$secGF,F.F*F.F.-%$tanGF," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "dsolve(ode03,x(t)): subs(_C1=C,%); \+
simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG*&,**&\"
\"\"F+-%$cosGF&!\"\"F+*&-%$sinGF&F+F,F.F+F'F.%\"CGF+F+,&-%$secGF&F+-%$
tanGF&F+F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,$*&,*!\"
\"\"\"\"-%$sinGF&F+*&F'F,-%$cosGF&F,F,*&%\"CGF,F0F,F+F,,&F,F,F-F,F+F+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
9 "Problem 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 39 "ode04:=diff(x(t),t)+tan(t)*x(t)=sec(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode04G/,&-%%diffG6$-%\"xG6#%\"tGF-
\"\"\"*&-%$tanGF,F.F*F.F.-%$secGF," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "dsolve(ode04,x(t)): subs(_C1=C,%); simplify(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&-%$cosGF&\"\"\"-%$ta
nGF&F,F,*&%\"CGF,F*F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%
\"tG,&-%$sinGF&\"\"\"*&%\"CGF+-%$cosGF&F+F+" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 5" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ode05:
=diff(y(x),x)=y(x)/(x+y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode
05G/-%%diffG6$-%\"yG6#%\"xGF,*&F)\"\"\",&F,F.F)F.!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "This equa
tion is not linear, but we can make it linear (by using the chain rule
)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "ode05b:=diff(x(y),y)=(x(y)+y)/y;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'ode05bG/-%%diffG6$-%\"xG6#%\"yGF,*&,&F)\"\"\"F,F/F/F
,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "dsolve(ode05b,x(y
)): subs(_C1=C,x(y)=x,%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG*&
,&-%#lnG6#%\"yG\"\"\"%\"CGF+F+F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 6" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 268 "A tank is full of brine \+
of concentration 2 g/L salt. Fresh water flows into the tank at 3 L/mi
n and the well-mixed solution is drawn off at the same rate. After 15 \+
minutes the concentration of brine in the outflow from the tank is 1.2
 g/L. Find the volume of the tank." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 80 "Let  Q  be the amount of salt in the tank
 and let  V  be the volume of the tank." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ode06:=diff(Q(t),t)=ri
*ei-(Q(t)/V)*ro;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode06G/-%%diffG
6$-%\"QG6#%\"tGF,,&*&%#riG\"\"\"%#eiGF0F0*&*&F)F0%#roGF0F0%\"VG!\"\"F6
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "init06:=Q(0)=2*V;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'init06G/-%\"QG6#\"\"!,$%\"VG\"\"#" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ode06b:=subs(ri=3,ro=3,ei
=0,ode06);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ode06bG/-%%diffG6$-%
\"QG6#%\"tGF,,$*&F)\"\"\"%\"VG!\"\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "soln06:=dsolve(\{ode06b,init06\},Q(t));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%'soln06G/-%\"QG6#%\"tG,$*&%\"VG\"\"\"-%$expG6#
,$*&F)F-F,!\"\"!\"$F-\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
56 "subs(t=15,rhs(soln06)); eqn06:=1.2*V=%; fsolve(eqn06,V);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*&%\"VG\"\"\"-%$expG6#,$*&F&F&F%!\"\"!#XF&
\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqn06G/,$%\"VG$\"#7!\"\",$
*&F'\"\"\"-%$expG6#,$*&F-F-F'F*!#XF-\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+]$o#4))!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 7" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ode07:=t*diff(x(t),t
)+4*x(t)=cos(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode07G/,&*&%\"t
G\"\"\"-%%diffG6$-%\"xG6#F(F(F)F)*&\"\"%F)F-F)F)-%$cosGF/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "init07:=x(Pi/2)=2/Pi;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%'init07G/-%\"xG6#,$%#PiG#\"\"\"\"\"#,$*&F,F,F*
!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dsolve(\{ode07,i
nit07\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG*&,,*&
)F'\"\"$\"\"\"-%$sinGF&F-F-*(F,F-)F'\"\"#F--%$cosGF&F-F-*&\"\"'F-F3F-!
\"\"*(F6F-F'F-F.F-F7*&F,F-%#PiGF-F-F-*$)F'\"\"%F-F7" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 8" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 46 "ode08:=diff(y(t),t)*cos(t)+sin(t)*y(t)=sin(t);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%&ode08G/,&*&-%%diffG6$-%\"yG6#%\"tGF.\"\"\"-%$cosG
F-F/F/*&-%$sinGF-F/F+F/F/F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "init08:=y(0)=Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'init08G/-%
\"yG6#\"\"!%#PiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dsolve(
\{ode08,init08\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%
\"tG,&\"\"\"F)*&-%$cosGF&F),&!\"\"F)%#PiGF)F)F)" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 9" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ode
09:=(1+x^2)*diff(y(x),x)+2*x*y(x)=3+3*x+2*x^2+x^3;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&ode09G/,&*&,&\"\"\"F)*$)%\"xG\"\"#F)F)F)-%%diffG6$
-%\"yG6#F,F,F)F)*(F-F)F,F)F1F)F),*\"\"$F)*&F6F)F,F)F)*&F-F)F+F)F)*$)F,
F6F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "dsolve(ode09,y(x)
): subs(_C1=C,%); simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"yG6#%\"xG*&,,F'\"\"$*&#F*\"\"#\"\"\")F'F-F.F.*&#F-F*F.)F'F*F.F.*&#F.
\"\"%F.)F'F5F.F.%\"CGF.F.,&F.F.*$F/F.F.!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&,,F'\"#O*&\"#=\"\"\")F'\"\"#F.F.*&\"
\")F.)F'\"\"$F.F.*&F4F.)F'\"\"%F.F.*&\"#7F.%\"CGF.F.F.,&F.F.*$F/F.F.!
\"\"#F.F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 10 "Problem 10" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "ode10:=(1+x)*diff(y(x),x)+y(x)=sin(
x)+cos(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode10G/,&*&,&\"\"\"F)
%\"xGF)F)-%%diffG6$-%\"yG6#F*F*F)F)F.F),&-%$sinGF0F)-%$cosGF0F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "dsolve(ode10,y(x)): subs(_C1
=C,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,(-%$sinGF&
\"\"\"-%$cosGF&!\"\"%\"CGF,F,,&F,F,F'F,F/" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Problem 11" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ode11:
=(1-x*y(x)^2)*diff(y(x),x)=y(x)^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%&ode11G/*&,&\"\"\"F(*&%\"xGF()-%\"yG6#F*\"\"#F(!\"\"F(-%%diffG6$F,F*
F(*$)F,\"\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "soln11:=d
solve(ode11,y(x)): subs(_C1=C,soln11[1]); subs(_C1=C,soln11[2]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&,&%\"CG\"\"\"*$-%%sq
rtG6#,&*$)F+\"\"#F,F,*&\"\"%F,F'F,!\"\"F,F,F,F'F7#F,F4" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&,&%\"CG\"\"\"*$-%%sqrtG6#,&*$)F+
\"\"#F,F,*&\"\"%F,F'F,!\"\"F,F7F,F'F7#F,F4" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The ode in this problem \+
may be regarded as linear  in x  (by means of the chain rule as in pro
blem 5):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "ode11b:=(1-x(y)*y^2)^(-1)*diff(x(y),y)=y^(-3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ode11bG/*&-%%diffG6$-%\"xG6#%\"yGF-
\"\"\",&F.F.*&F*F.)F-\"\"#F.!\"\"F3*&F.F.*$)F-\"\"$F.F3" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ode11c:=(1-x(y)*y^2)*ode11b;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ode11cG/-%%diffG6$-%\"xG6#%\"yGF,*&
,&\"\"\"F/*&F)F/)F,\"\"#F/!\"\"F/*$)F,\"\"$F/F3" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 42 "dsolve(ode11c,x(y)): subs(_C1=C,x(y)=x,%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG*&,&*&\"\"\"F(%\"yG!\"\"F*%\"CGF
(F(F)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 10 "Problem 12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ode12:=(t^2+1)*diff(x(t),t)+3*t^3*x
(t)=5*t^3*exp(-(3/2)*t^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode12
G/,&*&,&*$)%\"tG\"\"#\"\"\"F-F-F-F--%%diffG6$-%\"xG6#F+F+F-F-*(\"\"$F-
)F+F5F-F1F-F-,$*&F6F--%$expG6#,$F)#!\"$F,F-\"\"&" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 15 "init12:=x(0)=2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'init12G/-%\"xG6#\"\"!\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "dsolve(\{ode12,init12\},x(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&-%$expG6#,$*$)F'\"\"#\"\"\"#!\"$F0F1)
,&F.F1F1F1#\"\"$F0F1#\"#;F7*&#\"\"&F7F1*&,&F.F7F0F1F1F*F1F1!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "W
e can not always solve a linear ODE in terms of elementary functions. \+
If we make a small change to the ODE above we obtain an integral we ca
n not express in terms of elementary functions:" }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "ode12z:=(t
^2+1)*diff(x(t),t)+3*t^3*x(t)=5*t^3*exp(-(3/4)*t^2);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%'ode12zG/,&*&,&*$)%\"tG\"\"#\"\"\"F-F-F-F--%%diffG
6$-%\"xG6#F+F+F-F-*(\"\"$F-)F+F5F-F1F-F-,$*&F6F--%$expG6#,$F)#!\"$\"\"
%F-\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dsolve(\{ode12z
,init12\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG*(,&
-%$IntG6$,$*&*&)%\"uG\"\"$\"\"\"-%$expG6#,$*$)F1\"\"#F3#F2\"\"%F3F3*$)
,&F8F3F3F3#\"\"&F:F3!\"\"FA/F1;\"\"!F'F3F:F3F3-F56#,$*$)F'F:F3#!\"$F:F
3),&FIF3F3F3#F2F:F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
}{MARK "52 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }