{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "Helvetica" 1 14 128 0 0 1 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 24 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Helvetica " 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 " Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 }1 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 33 "Differential Equations \+ with Maple" }}{PARA 0 "" 0 "" {TEXT 256 20 "Mth 256 March 4 2001" }} {PARA 0 "" 0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 259 50 "Filename: 256winter2001_differential_equations.mws" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots ):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 238 "In this worksheet I give a few hints on using Maple to solve d ifferential equations. Maple can solve most of the differential equati ons encountered in Mth 256. Before we get to the differential equation s let's look at a few preliminaires." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 9 "Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 281 "Maple deals well with expressions. \+ Among the expressions that are of interest to us are equations. In Map le any expression, including an equation, may be assigned to a label. \+ Thus we can take a simple equations such as A=B and assign it to a lab el for ease in dealing with it. Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "eqn1:=A=B;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%eqn1G/%\"AG%\"BG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "Here eqn1 is the label ( or name) of the equation A=B. Sometimes we wish to extract the left or right (hand) side of an equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lhs(eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"AG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rhs(eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"BG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "There are devious, sometimes useful, ways to achieve the same thing. For examp le," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(eqn1,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"B G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 328 "Here we have taken the expression A and substituted A=B in it. Th e result is of course B. Note eqn1 is not altered in any way - it simp ly provides the specification for what to substitute for what. Note al so the subs() approach has the advantage that, unlike rhs(), it correc tly handles expressions containing several equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 183 "Another approach is t o use the assign() function. It assigns the right side of each equalit y in a list to the left side, that is the left side becomes a label fo r the right side. Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "assign(eqn1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A; B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"BG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"BG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "B:=6; A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG\" \"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "In case we want to use \+ A and B again let's unassign them:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "unassign('A','B'); " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "T he single quotes here prevent Maple from evaluating A and B. Otherwise we would be trying to unassign 6, which would produce an error." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 9 "Fun ctions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "A function in Maple is defined by the \"arrow\" notation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:= x->x^2+2*x-3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%) operatorG%&arrowGF(,(*$)9$\"\"#\"\"\"\"\"\"F/F0!\"$F2F(F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "A \+ suitable expression may be converted to a function by using unapply(). Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expr:=x^2+2*x-3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%exprG,(*$)%\"xG\"\"#\"\"\"\"\"\"F(F)!\"$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g:=unapply(expr,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(,(*$)9$\"\"#\" \"\"\"\"\"F/F0!\"$F2F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 129 "Functions are evaluated in the usual fam iliar way, whereas expressions are evaluated by using the substitute c ommand subs(). Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f(t); f(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"tG\"\"#\"\"\"\"\"\"F&F'!\"$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#X" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs (x=6,expr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#X" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "One thing to b e careful about is the x in the arrow notation above is local to the f unction we are defining, that is, it is just a dummy variable. Thus" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "h:=x->expr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGR6#%\"xG6\" 6$%)operatorG%&arrowGF(%%exprGF(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "does not define the same funct ion as f above, but instead defines the constant function whose value \+ is expr," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "h(t); h(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)% \"xG\"\"#\"\"\"\"\"\"F&F'!\"$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(* $)%\"xG\"\"#\"\"\"\"\"\"F&F'!\"$F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "It is best not to be too deviou s when defining functions. Errors like this can be hard to find!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 246 "Maple wo rks well with expressions, but is not quite as competent with function s. If one plans to use Maple to simplify complicated expressions, it i s best to leave them as expressions, even if habit makes functions fee l more natural to work with." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "is(f=g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "i s(f(x)=g(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Note FAIL does not mean false. It means Maple does not know the answer." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "The diff erentiation operator for functions is D(). The differentiation operato r for expressions is diff(). Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,&9$\"\"#F,\"\"\"F&F& F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(x),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"#F%\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Note that diff( ) is very easy to use for partial derivatives and for higher order der ivatives. Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "diff(arctan(y/x),x,y): simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*! \"\"F**$),&F&F+F,F+\"\"#F*!\"\"F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 504 "Note the use of Maple ditto opera tor % above. Be very careful! This operator refers to the previously e valuated expression (in time), not the previous expression on the work sheet. It makes a difference since one can move around and evaluate ex pressions in numerous places on the worksheet, so previous in time nee d not be the same as previous in location. The two are the same of cou rse if you restrict % to refer to an expression on the same line, as a bove. That is the safest way to use the % operator." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 22 "Differential Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 259 "We use the dsolve() command to so lve differential equations. It returns an expression (which may be emp ty) which contains equations giving the solution. There are also numer ous numeric versions which in a sense return a procedure for approxima ting a solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Here's an example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ode1:=diff(y(x),x)=x*sec(y(x ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode1G/-%%diffG6$-%\"yG6#%\"x GF,*&F,\"\"\"-%$secG6#F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "init1:=y(0)=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init1G/-%\"yG 6#\"\"!\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "soln1:=dsol ve(\{ode1,init1\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&soln1G/ -%\"yG6#%\"xG-%'arcsinG6#,&*$)F)\"\"#\"\"\"#\"\"\"F0-%$sinG6#F0F3" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "If one wants the solution expressed as a function y, that can be achieve d as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y:=unapply(rhs(soln1),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%'arcsinG6#,&* $)9$\"\"#\"\"\"#\"\"\"F3-%$sinG6#F3F6F(F(F(" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "The disadvantage here \+ is we have now assigned a value to y. We will not be able to use y aga in as a \"variable\" in a differential equation without first unassign ing it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "If you just want to plot the solution, or manipulate it in some o ther way, it is not necessary to convert it to a function. Thus" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot(rhs(soln1),x=-2/5..1/2,color=red,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6#7T7$$!1++++++ +S!#;$\"1I6uZ0OC9!#:7$$!1+++2G\">!RF*$\"1o54!>P)*R\"F-7$$!1+++8c#Q!QF* $\"1m%4XN#*)y8F-7$$!1+++r/[=PF*$\"1GySYNti8F-7$$!1+++F`8LOF*$\"1='yQ5c ![8F-7$$!1+++\\i8F-7$$!1+++TZ%zC$F*$\"1<;&Q*GR%H\"F -7$$!1+++H;jbIF*$\"1r%z_!z)HF\"F-7$$!1+++dFLxGF*$\"1STXx;Yb7F-7$$!1+++ -Yr#p#F*$\"1V$=]L$GR7F-7$$!1+++UAy,DF*$\"1zV;`'pVA\"F-7$$!1+++/AY6BF*$ \"1+%3$3E767F-7$$!1+++_Yp:@F*$\"1Z0!4Y0!*>\"F-7$$!1+++'[iK%>F*$\"1L>#> e)[*=\"F-7$$!1+++2Z9\\e>\"F*$\"1e2SU+6f6F-7$$! 1******>=\"*H**!#<$\"11]f\\vf`6F-7$$!1++++$)o6#)F`q$\"1+)paon(\\6F-7$$ !1+++qe)H@'F`q$\"1Z_!=Mai9\"F-7$$!1+++]bzVWF`q$\"14G`w9(R9\"F-7$$!1+++ qLr-DF`q$\"1r;CTeMU6F-7$$!1++++6WVl!#=$\"1/&315W;9\"F-7$$\"1+++5`Bu7F` q$\"1H(['zxyT6F-7$$\"1+++gKFXIF`q$\"1n#HjC3F9\"F-7$$\"1+++Spdb\\F`q$\" 1#eS;%GbW6F-7$$\"1******zb%)RpF`q$\"1m+gLjTZ6F-7$$\"1+++qs:n')F`q$\"14 #4T>42:\"F-7$$\"1+++@,F`5F*$\"1n8o7A7b6F-7$$\"1+++l!**fC\"F*$\"1,7$RvV 1;\"F-7$$\"1+++OnaM9F*$\"1)yox6Gq;\"F-7$$\"1+++.j(ph\"F*$\"1!***42D=F*$\"1N`uC7G$=\"F-7$$\"1+++X'R:+#F*$\"1*odXS*e#>\"F-7$ $\"1+++Q.(e>#F*$\"1'3#\\0Ez.7F-7$$\"1+++GG'>P#F*$\"1B^F8'p^@\"F-7$$\"1 +++L%yWc#F*$\"1fdLBy2H7F-7$$\"1+++81iXFF*$\"1%)>8B@tV7F-7$$\"1+++'Qm\\ $HF*$\"12p+&y24E\"F-7$$\"1+++)['3?JF*$\"1yus+v%)z7F-7$$\"1+++y+*QJ$F*$ \"1 " 0 "" {MPLTEXT 1 0 12 "unassign(y);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 32 "Sy stem of differential equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "Let's look at a simple system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ode 2:=diff(y(t),t)=3*x(t)-2*y(t), diff(x(t),t)=2*x(t)+3*y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode2G6$/-%%diffG6$-%\"yG6#%\"tGF-,&-%\"xG F,\"\"$F*!\"#/-F(6$F/F-,&F/\"\"#F*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "init2:=x(0)=3,y(0)=-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init2G6$/-%\"xG6#\"\"!\"\"$/-%\"yGF)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "soln2:=dsolve(\{ode2,init2\},\{x(t),y(t)\}) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&soln2G<$/-%\"xG6#%\"tG,*-%$exp G6#*&-%%sqrtG6#\"#8\"\"\"F*\"\"\"#\"\"$\"\"#*&F0F4-F-6#,$F/!\"\"F5#!\" $\"#E*&F0F4F,F5#F7F@F:F6/-%\"yGF),*F9#!#6F@FA#\"#6F@F,#F=F8F:FK" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "T he easiest way to extract the individual pieces is to use assign(). Al so assign() makes no assumptions about the order of the solutions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "assign(soln2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Note this assigns x(t) and y(t) as labels for t he appropriate expressions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%$expG6#*&-%%sqrtG6#\"#8\"\"\"%\"tG\"\"\"#\"\"$\"\"# *&F(F,-F%6#,$F'!\"\"F.#!\"$\"#E*&F(F,F$F.#F0F9F3F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,** &-%%sqrtG6#\"#8\"\"\"-%$expG6#,$*&F%F)%\"tG\"\"\"!\"\"F0#!#6\"#E*&F%F) -F+6#F.F0#\"#6F4F6#F1\"\"#F*F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 153 "It is improtant to realise that x an d y remain unassigned. If you want them to be functions corersponding \+ to the solutions you have to say so explicitly:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "x:=unapply( x(t),t); y:=unapply(y(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG R6#%\"tG6\"6$%)operatorG%&arrowGF(,*-%$expG6#*&-%%sqrtG6#\"#8\"\"\"9$ \"\"\"#\"\"$\"\"#*&F1F5-F.6#,$F0!\"\"F7#!\"$\"#E*&F1F5F-F7#F9FBF%\"yGR6#%\"tG6\"6$%)operatorG% &arrowGF(,**&-%%sqrtG6#\"#8\"\"\"-%$expG6#,$*&F.F29$\"\"\"!\"\"F9#!#6 \"#E*&F.F2-F46#F7F9#\"#6F=F?#F:\"\"#F3FCF(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Let's clean up befor e going to the next example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassign(x,y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "We do no t need to unassign x(t) and y(t). Those labels were clobbered when we \+ defined the functions x and y, since, after defining the functions, x( t) became the function x evaluated at t and y(t) became the function y evaluated at t. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 36 "Consider now a second order equation" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ode3 :=diff(y(x),x$2)+4*y(x)=sec(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% ode3G/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"F*\"\"%-%$secGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 171 "Note we used x$2 rather than x,x to indicate the second deriv ative. It's not a big deal here, but for higher order derivatives the \+ x$n notation has obvious advantages." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "soln3:=dsolve(ode3,y(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&soln3G/-%\"yG6#%\"xG,**&,&-%$si nGF(\"\"\"-%#lnG6#,&-%$secGF(F/-%$tanGF(F/#!\"\"\"\"#F/-F.6#,$F)F:F/F/ *&-%$cosGF(F/-F@F " 0 "" {MPLTEXT 1 0 24 "init3:=y(0)=A,D(y )(0)=B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init3G6$/-%\"yG6#\"\"!% \"AG/--%\"DG6#F(F)%\"BG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " soln3b:=dsolve(\{ode3,init3\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'soln3bG/-%\"yG6#%\"xG,**&,&-%$sinGF(\"\"\"-%#lnG6#,&-%$secGF(F/-% $tanGF(F/#!\"\"\"\"#F/-F.6#,$F)F:F/F/*&-%$cosGF(F/-F@F " 0 "" {MPLTEXT 1 0 45 "ode4:=diff(y(x),x )=x^2+x*y(x)+y(x)*tan(y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode 4G/-%%diffG6$-%\"yG6#%\"xGF,,(*$)F,\"\"#\"\"\"\"\"\"*&F,F2F)F2F2*&F)F1 -%$tanG6#F)F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "init4:=y (0)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init4G/-%\"yG6#\"\"!\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "soln4:=dsolve(\{ode4, init4\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&soln4G6\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Th e empty solution indicates Maple could not find a solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "s oln4n:=dsolve(\{ode4,init4\},y(x),numeric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'soln4nGR6#%(rkf45_xG6'%\"iG%(rkf45_sG%)outpointG%#r1 G%#r2G6#%aoCopyright~(c)~1993~by~the~University~of~Waterloo.~All~right s~reserved.G6\"C&>8&-%&evalfG6#9$@$52-%$absG6#,$F3!\"\"-F<6#,&&%,loc_c ontrolG6#\"\"#\"\"\"F3F?4-%'memberG6$&FD6#\"\"'<*F?FG!\"#FF$FF\"\"!$FG FR$F?FR$FPFRC%>FD-%%copyG6#=F06#;FG\"#EE\\[l;\"#CFR\"\"*\"%+5\"#DFR\"# 5FRFhnFR\"#6FR\"#7FR\"#8FR\"#9FR\"#:FR\"#;FRFGFG\"#FR\"\"%$FG!\")\"#?FS\"\"&Fjo\"#@FRFNFG\"#AFR\"\"($FG!\"*\"#BFR\" \")\"&++$>%'loc_y0G-FY6#=F06#;FGFGE\\[l\"FGFS>%'loc_y1G-FY6#=F0F[qE\\[ l!@$0F;FRC$>&FD6#FgoF3@%1%'DigitsG-%'evalhfG6#F\\rC$>8%-%*traperrorG6# -F^r6#-%=dsolve/numeric_solnall_rkf45G6,%&loc_FG-%$varG6#FD-F]s6#Fgp-F ]s6#F_q-F]s6#%'loc_F1G-F]s6#%'loc_F2G-F]s6#%'loc_F3G-F]s6#%'loc_F4G-F] s6#%'loc_F5G-F]s6#%)loc_workG@$/Fbr%*lasterrorGC%>8'-%+searchtextG6$.F ^r-%(convertG6$-%#opG6$FG7#Fbr%%nameG>8(-F\\u6$.%)hardwareGF_u@%50FjtF R0FhuFR-Fir6,F[sFDFgpF_qFesFhsF[tF^tFatFdt-%&ERRORG6#FbrFav7$/%\"xGF7- %$seqG6$/&%$ordG6#,&8$FGFGFG&Fgp6#Faw/FawF\\qF06%FDFgpF_qF0" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "T his looks mysterious, but fortunately some Maple functions such as ode plot() understand the returned data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "odeplot(soln4n,[x,y(x)],0. .0.1,thickness=3,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6#7T7$\"\"!$\"\"\"F(7$$\"+Fj\"3/#!#7$\"1% p`ov'>.5!#:7$$\"+aEj\"3%F.$\"1#**f>zIk+\"F17$$\"+\")*[C7'F.$\"1uSlAHq4 5F17$$\"+3`Ej\")F.$\"1)fyc*R,85F17$$\"+k\"3/-\"!#6$\"1eFt))[O;5F17$$\" +(z*[C7FD$\"1C647lv>5F17$$\"+I9dG9FD$\"1h(*R5)*=B5F17$$\"+jIlK;FD$\"19 ^akdmE5F17$$\"+'pMn$=FD$\"1i,o$R&=I5F17$$\"+Hj\"3/#FD$\"1)f&Rd(\\P.\"F 17$$\"+iz*[C#FD$\"1PG0e*ft.\"F17$$\"+&fz*[CFD$\"1rQHVr,T5F17$$\"+G71`E FD$\"1wYu3DsW5F17$$\"+hG9dGFD$\"1s2$4Ix%[5F17$$\"+%\\C71$FD$\"1&oD/#GG _5F17$$\"+FhIlKFD$\"16bDD/9c5F17$$\"+gxQpMFD$\"1kbeM:0g5F17$$\"+$RpMn$ FD$\"1h#=FjZ3@ @2\"F17$$\"+#H9dG%FD$\"1a>!Rxhi2\"F17$$\"+Dfz*[%FD$\"1^sFLTY!3\"F17$$ \"+ev(Qp%FD$\"1a>CY+t%3\"F17$$\"+\">fz*[FD$\"1W%>S\\h!*3\"F17$$\"+C3/- ^FD$\"13\"QIcgM4\"F17$$\"+dC71`FD$\"1)yVIXHz4\"F17$$\"+!4/-^&FD$\"1,x. '[qC5\"F17$$\"+BdG9dFD$\"1E]9;h326F17$$\"+ctO=fFD$\"1)eN0%*y<6\"F17$$ \"+*)*[C7'FD$\"1\"GP:r^l6\"F17$$\"+A1`EjFD$\"1Dc5]tS@6F17$$\"+bAhIlFD$ \"1]+%3'*[j7\"F17$$\"+))QpMnFD$\"1k*p([)z88\"F17$$\"+@bxQpFD$\"1tFLPzB<9\"F17$$\"+(yQpM(FD$\"1,0TWY/Z6F17$$\"+?/ -^vFD$\"1?-X5/Z_6F17$$\"+`?5bxFD$\"1fD1Md+e6F17$$\"+'o$=fzFD$\"1\"f&R9 clj6F17$$\"+>`Ej\")FD$\"1bDcZaUp6F17$$\"+_pMn$)FD$\"1'oYE2@`<\"F17$$\" +&eG9d)FD$\"1YL\"[#)[8=\"F17$$\"+=-^v()FD$\"1r2#pf:v=\"F17$$\"+^=fz*)F D$\"11ZN6*GQ>\"F17$$\"+%[tO=*FD$\"1)HUU+(H+7F17$$\"+<^v(Q*FD$\"1)QfZ#* Gp?\"F17$$\"+]n$=f*FD$\"19$yHlMP@\"F17$$\"+$Q=fz*FD$\"1/ZnS_s?7F17$$\" +-+++5!#5$\"1jj&4)H\"zA\"F1-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%*THIC KNESSG6#\"\"$" 1 2 0 1 3 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}} {EXCHG {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 12 "Hi gher Order" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Here is an example of a simple equation of fifth order" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "ode5:=diff(y(t),t$5)-4*diff(y(t),t$3)+3*diff(y(t),t$2)=0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode5G/,(-%%diffG6$-%\"yG6#%\"tG-%\" $G6$F-\"\"&\"\"\"-F(6$F*-F/6$F-\"\"$!\"%-F(6$F*-F/6$F-\"\"#F7\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "init5:=y(1)=2,D(y)(1)=0,(D@ @2)(y)(1)=-3,(D@@3)(y)(1)=2,(D@@4)(y)(1)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init5G6'/-%\"yG6#\"\"\"\"\"#/--%\"DG6#F(F)\"\"!/---% #@@G6$F/F+F0F)!\"$/---F66$F/\"\"$F0F)F+/---F66$F/\"\"%F0F)F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "soln5:=dsolve(\{ode5,init5\} ,y(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&soln5G/-%\"yG6#%\"tG,,# \"#P\"\"*\"\"\"F)#\"#8\"\"$*&-%$expGF(\"\"\"*$)-F46##F.\"\"#\"\"#F5!\" \"!#7*(,&\"$D$F.*$-%%sqrtG6#F0F5\"#%*F.-F46#,&F:F.FB#!\"\"F;F.-F46#,$* &,&FKF.FBF.F.F)F.F:F.#F.\"$<\"*(,&!$D$F.FBFFF.-F46#,&F:F.FBF:F.-F46#,$ *&,&F.F.FBF.F.F)F5FJF.#FKFR" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 139 "The solution is a bit difficult to inte rpret. Unless you really need the exact solution a floating point appr oximation might be preferable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "expr5:=evalf(rhs(soln5),6); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expr5G,,$\"'66T!\"&\"\"\"%\"tG$ \"'LLVF(-%$expG6#F*$!'c9WF(-F.6#,$F*$\"'y-8F($\"'?U:F(-F.6#,$F*$!'y-BF ($!'9!>\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 440 "Maple's numeric routines default to \"Digits\" decimal digits of precision. The default is 10, but it can be changed by assi gning a different number to Digits, for example, Digits:=20; The routi ne evalf() converts to floating point using the number of digits speci fied in Digits. Alternately one can specify the desired precision as t he second variable to evalf(). This has the advantage of not altering \+ the precision for the whole worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 14 "An LCR circuit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Consider a simple LCR circuit driven by a voltage " } {XPPEDIT 18 0 "4*cos(omega*t);" "6#*&\"\"%\"\"\"-%$cosG6#*&%&omegaGF%% \"tGF%F%" }{TEXT -1 9 " where " }{XPPEDIT 18 0 "0 < omega;" "6#2\"\" !%&omegaG" }{TEXT -1 108 " . Assume initially there is no charge on th e capacitor and no current flowing in the circuit. Then at time " } {XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT -1 56 " we switch on the source of electromotive force. We have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ode6:=L*diff(Q(t),t,t) +R*diff(Q(t),t)+(1/C)*Q(t)=4*cos(omega*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode6G/,(*&%\"LG\"\"\"-%%diffG6$-%\"QG6#%\"tG-%\"$G6$ F0\"\"#F)F)*&%\"RGF)-F+6$F-F0F)F)*&F-\"\"\"%\"CG!\"\"F),$-%$cosG6#*&%& omegaGF)F0F)\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "init6: =Q(0)=0,D(Q)(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init6G6$/-%\" QG6#\"\"!F*/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "Q(t);" "6#-%\"QG 6#%\"tG" }{TEXT -1 41 " is the charge on the capacitor at time " } {XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "Suppose the inductance is 1/4 \+ henry, the resistance is 100 ohms and the capacitance is 10^(-6) farad s. Then we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "ode6b:=subs(L=1/4,R=100,C=10^(-6),ode6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode6bG/,(-%%diffG6$-%\"QG6#%\"tG-% \"$G6$F-\"\"##\"\"\"\"\"%-F(6$F*F-\"$+\"F*\"(+++\",$-%$cosG6#*&%&omega GF3F-F3F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "soln6b:=dsolve (\{ode6b,init6\},Q(t)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 "The current " }{TEXT -1 80 "flowing in th e system is the rate of change of the charge on the capacitor. Thus" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "expr6:=diff(rhs(soln6b),t):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "We can think of the current " } {XPPEDIT 18 0 "J;" "6#%\"JG" }{TEXT -1 2 " " }{TEXT -1 18 "as a funct ion of " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 47 ". We have to unapply two variables in this case" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "J:=unapply(expr6,[omega,t]): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plot3d(J(omega,t),omeg a=0..6400,t=0..0.0059,axes=boxed,orientation=[35,55],thickness=2,numpo ints=1000);" }}{PARA 13 "" 1 "" {GLPLOT3D 462 396 396 {PLOTDATA 3 "6'- %%GRIDG6%;\"\"!$\"%+kF';F'$\"#f!\"%W(\\_o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