{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 263 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 264 "Tim es" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 265 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 268 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 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 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 }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 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 72 "A procedure for solving some 1st \+ order DE's by using Laplace transforms " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " } {TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 268 7 "DEsol.m" }{TEXT -1 32 " is required by t his worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 69 "A procedure for solving 1st or der linear DE's by Laplace transforms: " }{TEXT 0 9 "desolveLP" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "desolveLP: usage" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 260 2 " " } {TEXT -1 77 " desolveLP( de ) \n desolveLP( \{de,ic\} )\n de solveLP( \{de,ic\}, x(t) )" }{TEXT 261 1 "\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 10 "Parameters" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " de - " }{TEXT -1 92 " a first order differential equation with the derivative gi ven in the form diff(x(t),t)," }}{PARA 0 "" 0 "" {TEXT -1 94 " \+ (if t and x are the independent and dependent variable s respectively)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 10 " ic \+ - " }{TEXT -1 50 " an initial condition in the form x(t0) = x0. \+ " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }}{PARA 0 "" 0 "" {TEXT 267 11 "Description" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 9 "desolveLP" } {TEXT -1 65 " attempts to solve a first order linear differential of t he form " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+c*x = g(t);" "6#/,&*&%#dxG\"\"\"%#dtG! \"\"F'*&%\"cGF'%\"xGF'F'-%\"gG6#%\"tG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 28 "by using Laplace transforms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "If no initial condition \+ is given, a the solution is expressed in terms of an initial condition of the form x(0)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 93 "info=true/false\nThe option info=true causes intermediate steps in th e solution to be printed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 16 "How to activate: " }{TEXT 256 1 "\n" }{TEXT -1 155 "To make the procedure active open t he subsection, place the cursor anywhere after the prompt [ > and pre ss [Enter].\nYou can then close up the subsection. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "desolveLP: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9509 "desolveLP \+ := proc() \n local ff,vars,derivs,x,yx,y,df,dff,drv,e1,e2,prntflg, Options,\n de,ic,yy,xx,ee,lsic,rsic,initcond,x0,y0,startopts,pp,g x,\n L,ss,Gs,Hs,Ps,Yx,X,nvars,Ts,As,Ns,Ks,as,Tx,i;\n\n if nargs >0 then \n ff := args[1]\n else\n error \"at least one arg ument must be supplied\"\n end if;\n initcond := false;\n if typ e(ff,\{set(equation),list(equation)\}) and nops(ff)=2 then\n ff : = map(_u -> if has(_u,D) then convert(_u,diff) else _u end if,ff);\n \+ de := op(1,ff);\n ic := op(2,ff);\n if not has(de,diff) \+ then\n de := op(2,ff);\n ic := op(1,ff);\n end if ;\n initcond := true;\n elif type(ff,equation) then\n if h as(de,'D') then de := convert(de,'diff') else de := ff end if;\n els e\n error \"the 1st argument, %1, is invalid .. it should be an e quation or a set (or list) of 2 equations\",ff;\n end if;\n\n star topts := 2;\n if nargs>1 then\n ee := args[2];\n if type(e e,function) and nops(ee)=1 then\n yy := op(0,ee);\n xx := op(1,ee);\n if type(xx,name) and type(yy,name) then\n \+ startopts := 3;\n else\n error \"the 2nd arg ument, %1, has incorrect form for the dependent variable\",ee;\n \+ end if;\n end if;\n end if;\n\n prntflg := false;\n if n args>=startopts then\n Options:=[args[startopts..nargs]];\n \+ if not type(Options,list(equation)) then\n error \"each option al argument must be an equation\"\n end if;\n if hasoption(O ptions,'info','prntflg','Options') then \n if prntflg<>true th en prntflg := false end if;\n end if;\n if nops(Options)>0 t hen\n error \"%1 is not a valid option for %2\",op(1,Options), procname;\n end if;\n end if;\n\n derivs := indets(de,'specfu nc(anything,diff)');\n if derivs=\{\} then\n error \"the 1st ar gument, %1, is invalid .. it should be a differential equation or a se t (or list) containing a differential equation and an initial conditio n\",ff;\n end if;\n nvars := nops(indets(derivs,'name'));\n if n vars<>1 then\n if nvars=0 then\n error \"there is a probl em with the independent variable occurring in the derivative(s)\";\n \+ else\n error \"there should only be one independent variab le in the differential equation\"\n end if;\n end if;\n nvars := nops(indets(derivs,'anyfunc(name)'));\n if nvars<>1 then\n \+ if nvars=0 then\n error \"there is a problem with the dependen t variable occurring in the derivative(s)\"\n else\n erro r \"there should only be one dependent variable in the differential eq uation\"\n end if;\n end if;\n\n if nops(derivs)<>1 then\n \+ error \"there are too many derivatives in the differential equation .. note that the differential equation must be of order 1\"\n end i f;\n df := op(1,derivs);\n if type(df,function) and op(0,df)=diff \+ and nops(df)=2 then\n yx := op(1,df);\n if not type(yx,anyfu nc(name)) then\n error \"the 1st argument %1, in the derivativ e, %2, is invalid .. it should be the 'unknown' dependent variable\",y x,df;\n end if; \n x := op(2,df);\n if not type(x,name) then\n error \"the 2nd argument %1, in the derivative, %2, is invalid .. it should be the dependent variable\",x,df;\n end if; \n else\n error \"the derivative, %1, does not make sense\",df ;\n end if;\n\n if x<>s and eval(s)='s' then ss := s else ss := _s end if;\n \n y := op(0,yx);\n vars := indets(de,name);\n i f member(y,vars) then\n error \"%1 and %2 cannot both appear in t he differential equation\",yx,y;\n end if;\n if op(1,yx)<>x then\n error \"the derivative, %1, does not make sense\",df;\n end if ;\n\n if startopts=3 then \n if x<>xx or y<>yy then\n e rror \"cannot solve the differential equation for %1\",ee;\n end \+ if;\n end if;\n\n if initcond then\n lsic := lhs(ic);\n \+ if type(lsic,function) and op(0,lsic)=y and nops(lsic)=1 \n \+ and type(op(1,lsic),algebraic) then\n x0 := \+ op(1,lsic);\n if has(x0,\{x,y\}) then\n error \"ini tial condition must not involve %1 or %2\",x,y;\n end if;\n \+ else\n error \"initial condition is not decipherable\"\n \+ end if;\n rsic := rhs(ic);\n if type(rsic,algebraic) then \n y0 := rsic;\n if has(y0,\{x,y\}) then\n \+ error \"initial condition must not involve %1 or %2\",x,y;\n e nd if;\n else\n error \"initial condition is not decipher able\"\n end if;\n else\n x0 := 0;\n y0 := y(0);\n \+ end if;\n\n drv := solve(de,df);\n if nops([drv])=1 then\n e1 := subs(yx=y,drv);\n else\n error \"cannot obtain a unique exp ression for the derivative\"\n end if;\n\n e2 := expand(e1);\n i f not member(degree(e2,y),\{0,1\}) then\n error \"the DE is not l inear\"\n end if;\n pp := -traperror(simplify(coeff(e2,y,1)));\n \+ if fx=lasterror then\n error \"the DE is not linear\"\n end if ;\n if member(x,indets(pp)) then\n error \"the coefficients of \+ %1 and %2 in the DE must be independent of the variable %3\",Diff(yx, x),yx,x;\n end if;\n\n gx := combine(simplify(e2+pp*y),trig);\n \+ if member(y,indets(gx,name))\n then error \"the DE is not linear \"\n end if;\n\n if prntflg then\n print(``);\n print(`L inear DE . . `*Diff(yx,x)+pp*yx=gx);\n end if;\n\n if x0<>0 then \n gx := eval(subs(x=x+x0,gx));\n if prntflg then\n \+ print(``);\n print(`Change the independent variable to `*x=x-x 0,` so that `*y(0)=y0);\n print(`and solve . . `*Diff(yx,x)+pp *yx=gx);\n end if;\n end if; \n\n Gs := traperror(inttrans[l aplace](gx,x,ss));\n if Gs=lasterror or indets(Gs,'specfunc(anything ,laplace)')<>\{\} then\n error \"could not determine the Laplace \+ transform of %1\",gx;\n end if;\n\n if prntflg then\n print(` `);\n print(`Taking Laplace transforms gives . .`);\n print( ss*L*[yx]-y0+pp*L*[yx] = Gs);\n print(``);\n print(`so . . ` *L*[yx]*(ss+pp) = Gs+y0);\n end if;\n\n Hs := normal((Gs+y0)/(ss+ pp)); \n \n if prntflg then\n print(``);\n print(`Then \+ . . `*L*[yx] = Hs);\n end if;\n\n if has(indets(Hs,specfunc(anythi ng,'exp')),ss) then\n Ps := map(simplify,frontend(expand,[Hs])); \n if Ps<>Hs then\n Hs := Ps;\n if prntflg then\n print(``);\n print(``= Hs);\n end if;\n end if; \n if type(Hs,`+`) then\n Ps := 0; \n \+ Yx := 0;\n for i to nops(Hs) do\n Ts := op(i,Hs) ;\n As := select(has,[op(Ts)],'exp');\n if nops( As)=1 and type(op(As),specfunc('anything','exp')) \n \+ and not has(op(1,op(As))/ss,ss) then\n as := op(As)\n else\n as := 1\n end if;\n \+ Ns := Ts/as;\n if type(Ns,ratpoly('anything',ss)) then \n Ks := as*convert(Ns,parfrac,ss);\n Ps : = Ps + Ks: \n Ts := frontend(expand,[Ks]);\n \+ else\n Ps := Ps + Ts:\n end if;\n \+ Tx := traperror(inttrans[invlaplace](Ts,ss,x));\n if Tx=las terror or indets(Tx,'specfunc(anything,invlaplace)')<>\{\} then\n \+ error \"could not determine the inverse Laplace transform of %1\",Ts;\n end if;\n Yx := Yx + Tx;\n e nd do;\n if Ps<>Hs then\n if prntflg then\n \+ print(``);\n print(``= Ps);\n end if; \n end if;\n else\n As := select(has,[op(Hs)],'ex p');\n if nops(As)=1 and type(op(As),specfunc('anything','exp' )) \n and not has(op(1,op(As))/ss,ss) then\n \+ as := op(As)\n else\n as := 1\n end if; \n Ns := Hs/as;\n if type(Ns,ratpoly('anything',ss)) t hen \n Ts := frontend(expand,[as*convert(Ns,parfrac,ss)]); \n if Ts<>Hs then\n Hs := Ts;\n \+ if prntflg then\n print(``);\n prin t(``= Hs);\n end if;\n end if; \n end if;\n Yx := traperror(inttrans[invlaplace](Hs,ss,x));\n \+ if Yx=lasterror or indets(Yx,'specfunc(anything,invlaplace)')<>\{\} then\n error \"could not determine the inverse Laplace tra nsform of %1\",Hs;\n end if;\n end if;\n else\n if type(Hs,ratpoly('anything',ss)) then \n Ts := convert(Hs,parf rac,ss);\n if Ts<>Hs then\n Hs := Ts;\n \+ if prntflg then\n print(``);\n print(``= H s);\n end if;\n end if;\n else \n \+ Ts := map(simplify,frontend(expand,[Hs]));\n if Ts<>Hs then\n Hs := Ts;\n if prntflg then\n pri nt(``);\n print(``= Hs);\n end if;\n \+ end if;\n end if;\n Yx := traperror(inttrans[invlaplace](Hs, ss,x));\n if Yx=lasterror or indets(Yx,'specfunc(anything,invlapl ace)')<>\{\} then\n error \"could not determine the inverse La place transform of %1\",Hs;\n end if;\n end if;\n\n if has(Yx ,\{sinh,cosh\}) then Yx := simplify(convert(Yx,exp),exp) end if;\n i f x0<>0 then\n if prntflg then\n print(``);\n pri nt(`In terms of the shifted independent variable . . `);\n pri nt(yx=Yx);\n end if;\n Yx := eval(subs(x=x-x0,Yx));\n i f prntflg then\n print(``);\n print(`Returning to orig inal independent variable`,x=x+x0);\n print(yx=Yx);\n end if;\n end if;\n if prntflg then print(``) end if;\n\n return yx =Yx;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Examples are given in the following sections." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolveLP" }{TEXT -1 10 " : examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dt-3*y = exp(-2*t),y(0) = 1;" "6$/,&*&%#dyG\" \"\"%#dtG!\"\"F'*&\"\"$F'%\"yGF'F)-%$expG6#,$*&\"\"#F'%\"tGF'F)/-F,6# \"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "de := diff(y(t),t)-3*y(t)=exp(-2*t) ;\nic := y(0)=1;\ndesolveLP(\{de,ic\},y(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tGF-\"\"\"*&\"\"$F.F *F.!\"\"-%$expG6#,$*&\"\"#F.F-F.F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%0Linear~DE~.~.~~G\"\"\"-%%DiffG6 $-%\"yG6#%\"tGF.F'F'*&\"\"$F'F+F'!\"\"-%$expG6#,$*&\"\"#F'F.F'F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%DTaking~Laplace~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(%\"sG\"\"\"%\"LGF'7#-%\"yG6#%\"tGF'F'F'!\"\"*(\"\"$F'F(F'F) F'F.*&F'F',&F&F'\"\"#F'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\"\"\"%\"LGF&7#-%\"yG6#% \"tGF&,&%\"sGF&\"\"$!\"\"F&,&*&F&F&,&F.F&\"\"#F&F0F&F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then ~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&*(,&\"\"$F&%\"sGF&F&,&F0F&\"\"#F&! \"\",&F0F&F/F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"\"F'*&\"\"&F',&%\"sGF'\"\"#F'F'!\"\" F-*(\"\"'F'F)F-,&F+F'\"\"$F-F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,&*&#\"\"\"\"\"&F+- %$expG6#,$*&\"\"#F+F'F+!\"\"F+F3*&#\"\"'F,F+-F.6#,$*&\"\"$F+F'F+F+F+F+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG*&,&*&#\"\"\"\"\"&F,-%$expG6#,$*&F-F,F'F,!\"\"F,F3 #\"\"'F-F,F,-F/6#,$*&\"\"$F,F'F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 2" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dt+2*y = sin(3*t),y(0) = 0;" "6$/,&*&%#dyG\"\"\"%#dt G!\"\"F'*&\"\"#F'%\"yGF'F'-%$sinG6#*&\"\"$F'%\"tGF'/-F,6#\"\"!F6" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "de := diff(y(t),t)+2*y(t)=sin(3*t);\nic := y(0)= 0;\ndesolveLP(\{de,ic\},y(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tGF-\"\"\"*&\"\"#F.F*F.F.-%$sinG6# ,$*&\"\"$F.F-F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\" \"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%0Linear~DE~.~.~~G\"\"\"-%%DiffG6$-%\"yG6#%\"tGF.F 'F'*&\"\"#F'F+F'F'-%$sinG6#,$*&\"\"$F'F.F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Laplace ~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(%\"sG\" \"\"%\"LGF'7#-%\"yG6#%\"tGF'F'*(\"\"#F'F(F'F)F'F',$*&\"\"$F',&*$)F&F/F 'F'\"\"*F'!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,&%\" sGF&\"\"#F&F&,$*&\"\"$F&,&*$)F.F/F&F&\"\"*F&!\"\"F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~.~.~ G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,$*(\"\"$F&,&*$)%\"sG\"\"#F&F&\"\"*F&! \"\",&F3F&F4F&F6F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(\"\"$\"\"\"\"#8!\"\",&%\"sGF(\"\"#F(F*F (*(F)F*,&*&F'F(F,F(F*\"\"'F(F(,&*$)F,F-F(F(\"\"*F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#% \"tG,(*&#\"\"$\"#8\"\"\"-%$expG6#,$*&\"\"#F-F'F-!\"\"F-F-*&#F+F,F--%$c osG6#,$*&F+F-F'F-F-F-F4*&#F3F,F--%$sinGF9F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de, ic\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,(*&#\"\" $\"#8\"\"\"-%$expG6#,$*&\"\"#F-F'F-!\"\"F-F-*&#F+F,F--%$cosG6#,$*&F+F- F'F-F-F-F4*&#F3F,F--%$sinGF9F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 3" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dt+y = t*exp(-2*t),y(0) = 2;" "6$/,&*&%#dyG\"\"\"%#d tG!\"\"F'%\"yGF'*&%\"tGF'-%$expG6#,$*&\"\"#F'F,F'F)F'/-F*6#\"\"!F2" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "de := diff(y(t),t)+y(t)=t*exp(-2*t);\nic := y(0) =2;\ndesolveLP(\{de,ic\},y(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tGF-\"\"\"F*F.*&F-F.-%$ex pG6#,$*&\"\"#F.F-F.!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/- %\"yG6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,&*&%0Linear~DE~.~.~~G\"\"\"-%%DiffG6$-%\"yG6#% \"tGF.F'F'F+F'*&F.F'-%$expG6#,$*&\"\"#F'F.F'!\"\"F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Lapla ce~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(%\"sG \"\"\"%\"LGF'7#-%\"yG6#%\"tGF'F'\"\"#!\"\"*&F(F'F)F'F'*&F'F'*$),&F&F'F .F'F.F'F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,&%\"sGF&F&F &F&,&*&F&F&*$),&F.F&\"\"#F&F4F&!\"\"F&F4F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~.~.~G \"\"\"%\"LGF&7#-%\"yG6#%\"tGF&*(,(\"\"*F&*&\"\"#F&)%\"sGF1F&F&*&\"\")F &F3F&F&F&,&F3F&F1F&!\"#,&F3F&F&F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&\"\"\"F',&%\"sGF' \"\"#F'!\"\"F+*&F'F'*$)F(F*F'F+F+*&\"\"$F',&F)F'F'F'F+F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6# %\"tG,&*&\"\"$\"\"\"-%$expG6#,$F'!\"\"F+F+*&,&F+F0F'F0F+-F-6#,$*&\"\"# F+F'F+F0F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "dsolve(\{de,ic\},y(t));\nsimplify(expand(%));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG*&,(*&F'\"\"\"-%$expG6# ,$F'!\"\"F+F0F,F0\"\"$F+F+F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"tG,(-%$expG6#,$*&\"\"#\"\"\"F'F/!\"\"F0*&F'F/F)F/F0*&\"\"$F/- F*6#,$F'F0F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam ple 4" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dt+y = 1 +3*cos(2*t),y(0) = 4;" "6$/,&*&%#dyG\"\"\"%#dtG!\"\"F'%\"yGF',&F'F'*& \"\"$F'-%$cosG6#*&\"\"#F'%\"tGF'F'F'/-F*6#\"\"!\"\"%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "de := diff(y(t),t)+y(t)=1+3*cos(2*t);\nic := y(0)=4;\ndesolveLP( \{de,ic\},y(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/, &-%%diffG6$-%\"yG6#%\"tGF-\"\"\"F*F.,&F.F.*&\"\"$F.-%$cosG6#,$*&\"\"#F .F-F.F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\" \"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%0Linear~DE~.~.~~G\"\"\"-%%DiffG6$-%\"yG6#%\"tGF.F 'F'F+F',&F'F'*&\"\"$F'-%$cosG6#,$*&\"\"#F'F.F'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Lapl ace~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(%\"s G\"\"\"%\"LGF'7#-%\"yG6#%\"tGF'F'\"\"%!\"\"*&F(F'F)F'F',&*&F'F'F&F/F'* (\"\"$F'F&F',&*$)F&\"\"#F'F'F.F'F/F'" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\"\"\"%\"LGF&7#- %\"yG6#%\"tGF&,&%\"sGF&F&F&F&,(*&F&F&F.!\"\"F&*(\"\"$F&F.F&,&*$)F.\"\" #F&F&\"\"%F&F1F&F8F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF &,$*,\"\"%F&,**$)%\"sG\"\"#F&F&F&F&*$)F3\"\"$F&F&*&F/F&F3F&F&F&F3!\"\" ,&F1F&F/F&F9,&F3F&F&F&F9F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"\"&!\"\",&*&\"\"$\"\"\"%\"sG F,F,\"#7F,F,,&*$)F-\"\"#F,F,\"\"%F,F(F,*(F.F,F'F(,&F-F,F,F,F(F,*&F,F,F -F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,**&#\"\"$\"\"&\"\"\"-%$cosG6#,$*&\"\"#F- F'F-F-F-F-*&#\"\"'F,F--%$sinGF0F-F-*&#\"#7F,F--%$expG6#,$F'!\"\"F-F-F- F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,**&#\"\"$\"\"&\"\"\"-%$cosG6#,$*&\"\"#F-F'F-F-F-F -*&#\"\"'F,F--%$sinGF0F-F-*&#\"#7F,F--%$expG6#,$F'!\"\"F-F-F-F-" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dt+6*y = 3*t,y(1) = 2;" "6$/ ,&*&%#dyG\"\"\"%#dtG!\"\"F'*&\"\"'F'%\"yGF'F'*&\"\"$F'%\"tGF'/-F,6#F' \"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "de := diff(y(t),t)+6*y(t)=3*t;\nic := y(1 )=2;\ndesolveLP(\{de,ic\},y(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tGF-\"\"\"*&\"\"'F.F*F.F. ,$*&\"\"$F.F-F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\" \"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%0Linear~DE~.~.~~G\"\"\"-%%DiffG6$-%\"yG6#%\"tGF.F 'F'*&\"\"'F'F+F'F',$*&\"\"$F'F.F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/*&%DChange~the~independent~va riable~to~G\"\"\"%\"tGF&,&F&!\"\"F'F&/*&%*~so~that~GF&-%\"yG6#\"\"!F& \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%/and~solve~.~.~G\"\"\"- %%DiffG6$-%\"yG6#%\"tGF.F'F'*&\"\"'F'F+F'F',&*&\"\"$F'F.F'F'F3F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%DTaking~Laplace~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(%\"sG\"\"\"%\"LGF'7#-%\"yG6#%\"tGF'F'\"\"#!\"\"*(\"\"'F'F(F 'F)F'F',$*,\"\"$F'F&!\"#-%$expG6#F&F'-F76#,$F&F/F',&F&F'F'F'F'F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/**%(so~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,&%\"sGF&\"\"'F&F&,&*,\"\" $F&F.!\"#-%$expG6#F.F&-F56#,$F.!\"\"F&,&F.F&F&F&F&F&\"\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%* Then~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&*(,(**\"\"$F&-%$expG6#%\"sGF&- F26#,$F4!\"\"F&F4F&F&*(F0F&F1F&F5F&F&*&\"\"#F&)F4F;F&F&F&F4!\"#,&F4F& \"\"'F&F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"\"$\"\"\"%\"sG!\"\",&F)F(\"\"'F(F*F(*(F'F(F)! \"#F+F*F(*&\"\"#F(F+F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"#>\"\"\"\"#7!\"\",&%\"sGF(\" \"'F(F*F(*(\"\"&F(F)F*F,F*F(*&F(F(*&\"\"#F()F,F2F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%RIn~terms~ of~the~shifted~independent~variable~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,(*&#\"#>\"#7\"\"\"-%$expG6#,$*&\"\"'F-F' F-!\"\"F-F-#\"\"&F,F-*&\"\"#F4F'F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%KReturning~to~original~indep endent~variableG/%\"tG,&F%\"\"\"F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"yG6#%\"tG,(*&#\"#>\"#7\"\"\"-%$expG6#,&\"\"'F-*&F2F-F'F-!\"\"F- F-#F-F,F4*&\"\"#F4F'F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,(*&#\"#>\"#7\"\"\"-%$ex pG6#,&\"\"'F-*&F2F-F'F-!\"\"F-F-#F-F,F4*&\"\"#F4F'F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve( \{de,ic\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,(*& \"\"#!\"\"F'\"\"\"F,#F,\"#7F+*&#\"#>F.F,*&-%$expG6#,$*&\"\"'F,F'F,F+F, -F46#!\"'F+F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam ple 6" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dt+2*y = 12*t^3*exp(-t),y(0) = 0;" "6$/,&*&%#dyG\"\"\"%#dtG!\"\"F'*&\"\"#F'%\" yGF'F'*(\"#7F'*$%\"tG\"\"$F'-%$expG6#,$F0F)F'/-F,6#\"\"!F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "de := diff(y(t),t)+2*y(t)=12*t^3*exp(-t);\nic := y(0 )=0;\ndesolveLP(\{de,ic\},y(t),info=true);\nf := unapply(rhs(%),t);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tGF-\" \"\"*&\"\"#F.F*F.F.,$*(\"#7F.)F-\"\"$F.-%$expG6#,$F-!\"\"F.F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%0Line ar~DE~.~.~~G\"\"\"-%%DiffG6$-%\"yG6#%\"tGF.F'F'*&\"\"#F'F+F'F',$*(\"#7 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F&F&F&!\"%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*(%*Then~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,$*(\"#s F&,&%\"sGF&F&F&!\"%,&F1F&\"\"#F&!\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,,*&\"#s\"\"\",&%\"sG F(\"\"#F(!\"\"F(*&F'F(,&F*F(F(F(!\"$F,*&F'F(F.!\"#F(*&F'F(F.!\"%F(*&F' F(F.F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,&*&\"#s\"\"\"-%$expG6#,$*&\"\"#F+F'F+!\" \"F+F+*(\"#7F+,*\"\"'F2*&\"\"$F+)F'F1F+F2*&F6F+F'F+F+*$)F'F8F+F+F+-F-6 #,$F'F2F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)o peratorG%&arrowGF(,&*&\"#s\"\"\"-%$expG6#,$*&\"\"#F/9$F/!\"\"F/F/*(\"# 7F/,*\"\"'F7*&\"\"$F/)F6F5F/F7*&F;F/F6F/F/*$)F6F=F/F/F/-F16#,$F6F7F/F/ F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(t),t=0..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 437 195 195 {PLOTDATA 2 "6%-%'CURVESG6$7in7$$\"\"!F)F(7$$\"3emmm;arz@! #=$\"3V73C9ip;_!#?7$$\"3v***\\7y%*z7$F-$\"3Q:77W)>d(>!#>7$$\"3[LL$e9ui 2%F-$\"3G\\qt9%Q&*3&F67$$\"3z***\\(oMrU^F-$\"3o0lx=5'f8\"F-7$$\"3nmmm \"z_\"4iF-$\"3\")\\1t?RAF@F-7$$\"3Unmmm6m#G(F-$\"3)z\\qlo'fXNF-7$$\"39 ommT&phN)F-$\"3mIumzz-9aF-7$$\"3KLLe*=)H\\5!#<$\"3KNi)\\.Jo/\"FS7$$\"3 smm\"z/3uC\"FS$\"31gTw$R4vl\"FS7$$\"3ULLe*ot*\\8FS$\"3\\BN05=u;?FS7$$ \"3!****\\7LRDX\"FS$\"3O+'y+u\">)R#FS7$$\"3[LLekGhe:FS$\"3$)z9UWXw4GFS 7$$\"3%om;zR'ok;FS$\"3+=7R#HR@B$FS7$$\"3OLL3_(>/x\"FS$\"3a1\")z_%Qul$F S7$$\"33++D1J:w=FS$\"3)f#R35o&33%FS7$$\"3+n;HdG\"\\)>FS$\"3fo.=n-U3XFS 7$$\"3oLLL3En$4#FS$\"3suAztXKA\\FS7$$\"3_++Dc#o%*=#FS$\"3u@FSRZhr_FS7$ $\"3#pmmT!RE&G#FS$\"3:a&4jp6Ng&FS7$$\"3D+++D.&4]#FS$\"3AVbi%oG[F'FS7$$ \"3;+++vB_\"[%[\\(FS7$$\"3>+]7.%Q%GKFS$\"39E8$efI=g(FS7$$\"3 _LLL347TLFS$\"3o>2b1yHuwFS7$$\"3#QLL3xxlV$FS$\"3>>M,AQv4xFS7$$\"3nLLLL Y.KNFS$\"3bvlGY<^AxFS7$$\"3mm;HdO2VOFS$\"3-48!\\P6.r(FS7$$\"33++D\"o7T v$FS$\"3e$>WK$y(3n(FS7$$\"3km;HK5S_QFS$\"3$zx_Wf')[h(FS7$$\"3?LLL$Q*o] RFS$\"3cFu!R&fhSvFS7$$\"3m++D\"=lj;%FS$\"3xWmn$\\R3K(FS7$$\"3S++vV&R

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1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,*\"\"\"F)-%\"HG6#,&F'F)\"\"#!\"\"F/*&F*F )-%$expG6#,&F.F)F'F/F)F)-F26#,$F'F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 10 "Example 10" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dt+y = t*H(t)+(2-2*t)*H(t-1)+(t-2)*H(t-2),y(0) = 0; " "6$/,&*&%#dyG\"\"\"%#dtG!\"\"F'%\"yGF',(*&%\"tGF'-%\"HG6#F-F'F'*&,& \"\"#F'*&F3F'F-F'F)F'-F/6#,&F-F'F'F)F'F'*&,&F-F'F3F)F'-F/6#,&F-F'F3F)F 'F'/-F*6#\"\"!F@" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 39 "where H is the Heaviside step function." }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "alias(H=Heaviside):\nf \+ := t -> t*H(t)+(2-2*t)*H(t-1)+(t-2)*H(t-2):\nde := diff(y(t),t)+y(t)=f (t);\nic := y(0)=0;\ndesolveLP(\{de,ic\},y(t),info=true);\ng := unappl y(rhs(%),t):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-% \"yG6#%\"tGF-\"\"\"F*F.,(*&F-F.-%\"HGF,F.F.*&,&\"\"#F.*&F5F.F-F.!\"\"F .-F26#,&F.F7F-F.F.F.*&-F26#,&F-F.F5F7F.F>F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%0Linear~DE~.~.~~G\"\" \"-%%DiffG6$-%\"yG6#%\"tGF.F'F'F+F',,*&F.F'-%\"HGF-F'F'*&\"\"#F'-F26#, &F'!\"\"F.F'F'F'*(F4F'F.F'F5F'F8*&F.F'-F26#,&F.F'F4F8F'F'*&F4F'F;F'F8 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Laplace~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(%\"sG\"\"\"%\"LGF'7#-%\"yG6#%\"tGF'F'*&F(F'F)F'F', (*&F'F'*$)F&\"\"#F'!\"\"F'*(F3F'-%$expG6#,$F&F4F'F&!\"#F4*&-F76#,$*&F3 F'F&F'F4F'F&F:F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,&%\" 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"" 0 "" {TEXT -1 10 "Example 11 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt+y = 1+Sum(( -1)^k*2*H(t-k),k=1..4)-H(t-5),y(0)=0" "6$/,&*&%#dyG\"\"\"%#dtG!\"\"F'% \"yGF',(F'F'-%$SumG6$*(),$F'F)%\"kGF'\"\"#F'-%\"HG6#,&%\"tGF'F2F)F'/F2 ;F'\"\"%F'-F56#,&F8F'\"\"&F)F)/-F*6#\"\"!FC" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "where H is the Heaviside step function." }} {PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "alias(H=Heaviside):\nf := t -> 1+sum((-1)^k*2*H(t-k),k=1..4)- H(t-5):\nde := diff(y(t),t)+y(t)=f(t);\nic := y(0)=0;\ndesolveLP(\{de, ic\},y(t),info=true);\ng := unapply(rhs(%),t):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tGF-\"\"\"F*F.,.F.F.*&\" \"#F.-%\"HG6#,&F.!\"\"F-F.F.F6*&F1F.-F36#,&F-F.F1F6F.F.*&F1F.-F36#,&F- F.\"\"$F6F.F6*&F1F.-F36#,&F-F.\"\"%F6F.F.-F36#,&F-F.\"\"&F6F6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%0Line ar~DE~.~.~~G\"\"\"-%%DiffG6$-%\"yG6#%\"tGF.F'F'F+F',.F'F'*&\"\"#F'-%\" HG6#,&F'!\"\"F.F'F'F6*&F1F'-F36#,&F.F'F1F6F'F'*&F1F'-F36#,&F.F'\"\"$F6 F'F6*&F1F'-F36#,&F.F'\"\"%F6F'F'-F36#,&F.F'\"\"&F6F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Lapl ace~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(%\"s G\"\"\"%\"LGF'7#-%\"yG6#%\"tGF'F'*&F(F'F)F'F',.*&F'F'F&!\"\"F'*(\"\"#F '-%$expG6#,$F&F1F'F&F1F1*(F3F'-F56#,$*&F3F'F&F'F1F'F&F1F'*(F3F'-F56#,$ *&\"\"$F'F&F'F1F'F&F1F1*(F3F'-F56#,$*&\"\"%F'F&F'F1F'F&F1F'*&-F56#,$*& \"\"&F'F&F'F1F'F&F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&, &%\"sGF&F&F&F&,.*&F&F&F.!\"\"F&*(\"\"#F&-%$expG6#,$F.F1F&F.F1F1*(F3F&- F56#,$*&F3F&F.F&F1F&F.F1F&*(F3F&-F56#,$*&\"\"$F&F.F&F1F&F.F1F1*(F3F&-F 56#,$*&\"\"%F&F.F&F1F&F.F1F&*&-F56#,$*&\"\"&F&F.F&F1F&F.F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%* Then~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,$*(,.F&!\"\"*&\"\"#F&-%$expG6 #,$%\"sGF0F&F&*&F2F&-F46#,$*&F2F&F7F&F0F&F0*&F2F&-F46#,$*&\"\"$F&F7F&F 0F&F&*&F2F&-F46#,$*&\"\"%F&F7F&F0F&F0-F46#,$*&\"\"&F&F7F&F0F&F&F7F0,&F 7F&F&F&F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,.*&\"\"\"F'*&%\"sGF',&F)F'F'F'F'!\"\"F'**\"\"#F' F)F+F*F+-%$expG6#,$F)F+F'F+**F-F'F)F+F*F+-F/6#,$*&F-F'F)F'F+F'F'**F-F' F)F+F*F+-F/6#,$*&\"\"$F'F)F'F+F'F+**F-F'F)F+F*F+-F/6#,$*&\"\"%F'F)F'F+ F'F'*(F)F+F*F+-F/6#,$*&\"\"&F'F)F'F+F'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,0*&\"\"\"F',&%\"sGF' F'F'!\"\"F**&F'F'F)F*F'*&-%$expG6#,$F)F*F',&*&\"\"#F'F(F*F'*&F3F'F)F*F *F'F'*&-F.6#,$*&F3F'F)F'F*F',&*&F3F'F(F*F**&F3F'F)F*F'F'F'*&-F.6#,$*& \"\"$F'F)F'F*F'F1F'F'*&-F.6#,$*&\"\"%F'F)F'F*F'F:F'F'*&-F.6#,$*&\"\"&F 'F)F'F*F',&F&F'F+F*F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,0-%$expG6#,$F'!\"\"F-\" 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" 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 12" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt+2*y = 4/Pi;" "6 #/,&*&%#dyG\"\"\"%#dtG!\"\"F'*&\"\"#F'%\"yGF'F'*&\"\"%F'%#PiGF)" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``(sin(Pi*t)+sin(3*Pi*t)/3+sin(5*Pi*t)/ 5+sin(7*Pi*t)/7+sin(9*Pi*t)/9+sin(11*Pi*t)/11),y(0) = 0;" "6$-%!G6#,.- %$sinG6#*&%#PiG\"\"\"%\"tGF,F,*&-F(6#*(\"\"$F,F+F,F-F,F,F2!\"\"F,*&-F( 6#*(\"\"&F,F+F,F-F,F,F8F3F,*&-F(6#*(\"\"(F,F+F,F-F,F,F=F3F,*&-F(6#*(\" \"*F,F+F,F-F,F,FBF3F,*&-F(6#*(\"#6F,F+F,F-F,F,FGF3F,/-%\"yG6#\"\"!FL" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 217 "f := t -> 4/Pi*(sin(Pi*t)+sin(3*Pi*t)/3+sin (5*Pi*t)/5+sin(7*Pi*t)/7+\n sin(9*Pi*t)/9+sin(11*Pi*t)/11):\nde : = diff(y(t),t)+2*y(t)=f(t);\nic := y(0)=0;\ndesolveLP(\{de,ic\},y(t),i nfo=true);\ng := unapply(sort(rhs(%)),t):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tGF-\"\"\"*&\"\"#F.F*F.F. ,$*(\"\"%F.%#PiG!\"\",.-%$sinG6#*&F-F.F4F.F.*&#F.\"\"$F.-F86#,$*(F=F.F -F.F4F.F.F.F.*&#F.\"\"&F.-F86#,$*(FDF.F-F.F4F.F.F.F.*&#F.\"\"(F.-F86#, $*(FKF.F-F.F4F.F.F.F.*&#F.\"\"*F.-F86#,$*(FRF.F-F.F4F.F.F.F.*&#F.\"#6F .-F86#,$*(FYF.F-F.F4F.F.F.F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,&*&%0Linear~DE~.~.~~G\"\"\"-%%DiffG6$-%\"y G6#%\"tGF.F'F'*&\"\"#F'F+F'F',$*&#F'\"%lMF'*&,.*&\"&gQ\"F'-%$sinG6#*&F .F'%#PiGF'F'F'*&\"%?YF'-F:6#,$*(\"\"$F'F.F'F=F'F'F'F'*&\"%sFF'-F:6#,$* (\"\"&F'F.F'F=F'F'F'F'*&\"%!)>F'-F:6#,$*(\"\"(F'F.F'F=F'F'F'F'*&\"%S:F '-F:6#,$*(\"\"*F'F.F'F=F'F'F'F'*&\"%g7F'-F:6#,$*(\"#6F'F.F'F=F'F'F'F'F 'F=!\"\"F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%DTaking~Laplace~transforms~gives~.~.G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,&*(%\"sG\"\"\"%\"LGF'7#-%\"yG6#%\"tGF'F'*(\"\" #F'F(F'F)F'F',$*(\"%lM!\"\"%#PiGF3,.*(\"&gQ\"F'F4F',&*$)F&F/F'F'*$)F4F 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