{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 "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{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 "" -1 263 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 265 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "Purple Emphasis" -1 266 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 267 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 270 "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 "Times" 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 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 71 "A procedure for solving some 2nd \+ 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 270 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 2nd or der linear DE's by Laplace transforms: " }{TEXT 0 9 "desolveL2" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "desolveL2: usage" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 17 " desolveC C( de )" }}{PARA 0 "" 0 "" {TEXT -1 52 " desolveCC( de ,y(x) )\n des olveCC( \{de,cnstrts \}) " }}{PARA 0 "" 0 "" {TEXT -1 34 " desolveCC( \{de,cnstrts \},y(x) ) " }{TEXT 262 1 "\n" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 268 10 "Parameters" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 9 " de - " }{TEXT -1 70 " a 2nd order linear differential equation with constant coefficie nts," }}{PARA 0 "" 0 "" {TEXT -1 65 " that is, one \+ which can be written in the form" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "a ;" "6#%\"aG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+b;" "6#,&*( %\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(%\"bGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+c*y = f(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&% \"cGF'%\"yGF'F'-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 117 " with the first and 2nd order derivati ves entered as diff(y(x),x) and diff(y(x),x$2) respectively." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" } {TEXT -1 82 ": The dependent variable,say y, must be entered as y(x) e verywhere in the equation" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 23 12 "cnstrts - " }{TEXT 263 86 "two constraints involving the depende nt variable, or its derivative, given in the form" }}{PARA 0 "" 0 "" {TEXT 265 53 " y(a)=b or D(y)(a)=b " } }{PARA 0 "" 0 "" {TEXT -1 67 " so that \{de,cnstrts\} is a set of three equations." }}{PARA 0 "" 0 "" {TEXT 264 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 11 "Descript ion" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 14 "The procedure " }{TEXT 0 9 "desolveL2" }{TEXT -1 116 " \+ attempts to solve a 2nd order linear differential equation with const ant coefficients by using Laplace transforms." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 261 8 "Options:" }{TEXT -1 1 " \n" }}{PARA 0 "" 0 "" {TEXT -1 80 "info=true/false\nWith the option \" info=true\" the steps in the solution are shown." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 155 "To ma ke the procedure active, open the subsection, place the cursor anywher e after the prompt [ > and press [Enter].\nYou can then close up the \+ subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "desolveL2: impleme ntation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15447 "desolveL2 := proc()\n local ff,de,ic0,ic1,x0,y0, x1,y1,derivs,df,df1,df2,df3,dff,\n pm1,pm2,x,y,yx,x2,x3,yx2,pol,vars ,d0,d1,d2,c0,c1,c2,c3,\n tt,lsic,initcond,startopts,ee,xx,yy,prntflg ,Options,t0,\n t1,t2,ratfact,algfact,cd,ss,Gs,Hs,Ps,Yx,nvars,order, \n As,Ts,as,Ns,Ks,Tx,i;\n\n# extract common denom of rational factor s from alg expr or list\nratfact := proc(ff)\n local fact,rfact,term s,rterms,i;\n if op(0,ff)=`list` then\n rterms := NULL;\n \+ for i to nops(ff) do\n rterms := rterms,ratfact(ff[i])\n \+ end do;\n return ilcm(rterms);\n elif op(0,ff)=`*` then\n \+ fact := [op(ff)];\n rfact := NULL;\n for i to nops(fact) do \n if type(fact[i],rational) then \n rfact := rfact ,fact[i]\n end if;\n end do;\n if nops([rfact])=1 th en return denom(abs(rfact))\n else return 1 end if;\n elif op(0 ,ff)=`+` then\n terms := [op(ff)];\n rterms := NULL;\n \+ for i to nops(terms) do\n rterms := rterms,ratfact(terms[i])\n end do;\n return ilcm(rterms);\n elif type(ff,rational) t hen\n return denom(abs(ff))\n else \n return 1\n end if; \nend proc; # of ratfact\n\n# extract algebraic common denom from alg \+ expr or list\nalgfact := proc(ff,x)\n local i,fact,build,C,ee,t;\n \+ if op(0,ff)=`list` then\n C := table();\n ee := add(C[i]*ff [i],i=1..nops(ff));\n else\n ee := ff;\n end if; \n fact \+ := factor(ee);\n build := 1; \n if op(0,fact)=`*` then\n fo r i to nops(fact) do\n t := op(i,fact);\n if numer(t)= 1 and not has(denom(t),x) then\n build := build*denom(t);\n end if;\n end do;\n end if;\n build;\nend proc;\n\n \+ # start of main procedure\n if nargs>0 then \n ff := args[1] \n else\n error \"at least one argument must be supplied\"\n \+ end if;\n initcond := false;\n if type(ff,\{set(equation),list(equ ation)\}) and nops(ff)=3 then\n ff := map(_u -> if has(_u,D@@2) t hen convert(_u,diff) else _u end if,ff);\n de := op(1,ff);\n \+ ic0 := op(2,ff);\n ic1 := op(3,ff);\n if not has(de,diff) t hen\n de := op(2,ff);\n ic0 := op(1,ff);\n end if ;\n if not has(de,diff) then\n de := op(3,ff);\n \+ ic1 := op(2,ff);\n end if;\n initcond := true;\n elif type (ff,equation) then\n de := ff;\n else\n error \"the 1st ar gument, %1, is invalid .. it should be an equation or a set (or list) \+ of 3 equations\",ff;\n end if;\n\n startopts := 2;\n if nargs>1 \+ then\n ee := args[2];\n if type(ee,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 argument, %1, has incorrect fo rm for the dependent variable\",ee;\n end if;\n end if;\n end if;\n\n prntflg := false;\n if nargs>=startopts then\n \+ Options:=[args[startopts..nargs]];\n if not type(Options,list(eq uation)) then\n error \"each optional argument must be an equa tion\"\n end if;\n if hasoption(Options,'info','prntflg','Op tions') then \n if prntflg<>true then prntflg := false end if; \n end if;\n if nops(Options)>0 then\n error \"%1 is not a valid option for %2\",op(1,Options),procname;\n end if;\n \+ end if;\n \n # Check out the derivatives in the DE.\n derivs : = indets(de,'specfunc(anything,diff)');\n if derivs=\{\} then\n \+ error \"the 1st argument, %1, is invalid .. it should be a differenti al equation or a set (or list) containing a differential equation and \+ two initial conditions\",ff;\n end if;\n nvars := nops(indets(deri vs,name));\n if nvars<>1 then\n if nvars=0 then\n error \"there is a problem with the independent variable occurring in the d erivative(s)\";\n else\n error \"there should only be one independent variable in the differential equation\"\n end if;\n \+ end if;\n nvars := nops(indets(derivs,anyfunc(name)));\n if nvar s<>1 then\n if nvars=0 then\n error \"there is a problem \+ with the dependent variable occurring in the derivative(s)\"\n el se\n error \"there should only be one dependent variable in th e differential equation\"\n end if;\n end if;\n\n order := no ps(derivs);\n if order=1 then\n error \"the differential equat ion should have order 2\" \n elif order>2 then\n error \"there \+ are too many derivatives in the differential equation .. note that the differential equation should have order 2\"\n end if;\n\n (df2,df 1) := selectremove(_U->has([op(_U)],diff),derivs);\n if nops(df2)<>1 or nops(df1)<>1 then \n error \"the derivatives, %1, do not make sense\",derivs;\n end if; \n (df2,df1) := (op(df2),op(df1));\n\n \+ # Get the arguments in the derivatives.\n if type(df1,function) an d op(0,df1)=diff and nops(df1)=2 then\n yx := op(1,df1);\n i f not type(yx,anyfunc(name)) then\n error \"the 1st argument % 1, in the derivative, %2, is invalid .. it should be the 'unknown' dep endent variable\",yx,df1;\n end if; \n x := op(2,df1);\n \+ if not type(x,name) then\n error \"the 2nd argument %1, in t he derivative, %2, is invalid .. it should be the dependent variable\" ,x,df1;\n end if; \n else\n error \"the derivative, %1, do es not make sense\",df1;\n end if;\n\n if type(df2,function) and n ops(df2)=2 and op(0,df2)='diff' then\n (df3,x3) := selectremove(h as,\{op(df2)\},diff);\n if nops(df3)<>1 or nops(x3)<>1 then \n \+ error \"the derivative, %1, does not make sense\",df2;\n en d if;\n (df3,x3) := (op(df3),op(x3));\n if type(df3,function ) and nops(df3)=2 and op(0,df3)='diff' then\n yx2 := op(1,df3) ;\n if not type(yx2,anyfunc(name)) then\n error \"t he 1st argument %1, in the derivative, %2, is invalid .. it should be \+ the 'unknown' dependent variable\",yx2,df3;\n end if; \n \+ x2 := op(2,df2);\n if not type(x2,name) then\n e rror \"the 2nd argument %1, in the derivative, %2, is invalid .. it sh ould be the dependent variable\",x2,df3;\n end if; \n \+ if not x2=x3 then\n error \"the 2nd arguments, %1 and %2 in the derivatives %3 and %4 should be the same\",x2,x3,df2,df3;\n \+ end if;\n else\n error \"the derivative, %1, does not \+ make sense\",df3;\n end if\n else\n error \"the derivative , %1, does not make sense\",df2;\n end if;\n\n # Arguments in the \+ 2 derivatives must be the same.\n if x2<>x or yx2<>yx then\n er ror \"the differential equation contains inconsistent arguments\"\n \+ end if;\n\n y := op(0,yx);\n vars := indets(de,name);\n if membe r(y,vars) then\n error \"%1 and %2 cannot both appear in the diff erential equation\",yx,y;\n end if;\n if op(1,yx)<>x then\n e rror \"the derivatives do not make sense\"\n end if;\n\n if starto pts=3 then \n if x<>xx or y<>yy then\n error \"cannot sol ve the differential equation for %1\",ee;\n end if;\n end if;\n \n if x<>s and eval(s)='s' then ss := s else ss := _s end if;\n\n \+ # Form a polynomial by substituting for the derivatives.\023\n pol:= subs(yx=d0,subs(diff(yx,x)=d1,\n subs(diff(yx,x$2)=d 2,de)));\n pol := d2-expand(rhs(isolate(pol,d2)));\n if degree(pol ,d2)<>1 or not member(degree(pol,d1),\{0,1\})\n \+ or not member(degree(pol,d0),\{0,1\}) then\n error \"the DE is no t linear\"\n end if;\n \n # Coefficients of DE are polynomial co efficients.\n c2 := traperror(coeff(pol,d1,1));\n if c2=lasterror \+ or member(d0,indets(c2,name)) or\n member(d2,indets(c2,name)) the n \n error \"the DE is not linear\"\n end if;\n c1 := traperr or(coeff(pol,d0,1));\n if c1=lasterror or member(d1,indets(c1,name)) \n or member(d2,indets(c1,name)) then\n error \"the DE is no t linear\"\n end if;\n c0 := traperror(coeff(pol,d1,0));\n if c0 =lasterror then error \"the DE is not linear\" end if;\n c0 := simpl ify(d2+c1*d0-c0);\n if member(d0,indets(c0,name)) or member(d2,indet s(c0,name)) then\n error \"the DE is not linear\"\n end if;\n \+ c2 := simplify(c2);\n c1 := simplify(c1);\n \n if member(x,indets (c2,name)) or member(x,indets(c1,name)) then\n error \"the coeffi cients must be independent of the variable %1\",x;\n end if;\n\n i f initcond then\n # Get the initial conditions.\n lsic := lh s(ic0);\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 \"the initial conditions must not involve %1 or %2\",x,y;\n \+ end if;\n y0 := rhs(ic0);\n t0 := 0; # flag fo r y coord or derivative\n if has(y0,\{x,y\}) then\n \+ error \"the initial conditions must not involve %1 or %2\",x,y;\n \+ end if;\n elif type(lsic,function) and op(0,lsic)=D(y) and \+ nops(lsic)=1 \n and type(op(1,lsic),algebr aic) then\n x0 := op(1,lsic);\n if has(x0,\{x,y\}) the n\n error \"the initial conditions must not involve %1 or % 2\",x,y;\n end if;\n y0 := rhs(ic0);\n t0 := 1 ; # flag for y coord or derivative\n if has(y0,\{x,y\}) then\n error \"the initial conditions must not involve %1 or %2\" ,x,y;\n end if;\n else\n error \"initial conditio n is not decipherable\"\n end if;\n\n lsic := lhs(ic1);\n \+ if type(lsic,function) and op(0,lsic)=y and nops(lsic)=1 \n \+ and type(op(1,lsic),algebraic) then\n x1 \+ := op(1,lsic);\n if has(x1,\{x,y\}) then\n error \" the initial conditions must not involve %1 or %2\",x,y;\n end \+ if;\n y1 := rhs(ic1);\n t1 := 0; # flag for y coord or derivative\n if has(y1,\{x,y\}) then\n error \"the initial conditions must not involve %1 or %2\",x,y;\n end if; \n elif type(lsic,function) and op(0,lsic)=D(y) and nops(lsic)=1 \+ \n and type(op(1,lsic),algebraic) then\n \+ x1 := op(1,lsic);\n if has(x1,\{x,y\}) then\n \+ error \"the initial conditions must not involve %1 or %2\",x,y;\n \+ end if;\n y1 := rhs(ic1);\n t1 := 1; # flag for \+ y coord or derivative\n if has(y1,\{x,y\}) then\n e rror \"the initial conditions must not involve %1 or %2\",x,y;\n \+ end if;\n else\n error \"initial condition is not deci pherable\"\n end if;\n\n if x0<>x1 then \n error \"i nitial conditions must supply a value and derivative at a single value of %1\",x;\n end if;\n if t0=t1 then\n error \"impo ssible initial conditions\"\n end if;\n if t0=1 then # swap \+ over\n tt := y1; y1 := y0; y0 := tt;\n end if;\n else\n x0 := 0;\n y0 := y(0);\n y1 := D(y)(0);\n end if;\n \n c3 := algfact([c0,c1,c2],x);\n c2 := simplify(c2*c3);\n c1 := simplify(c1*c3);\n c0 := simplify(c0*c3);\n cd := ratfact([c0,c1, c2,c3]);\n c3 := simplify(c3*cd);\n c2 := simplify(c2*cd);\n c1 \+ := simplify(c1*cd);\n c0 := combine(simplify(c0*cd),trig);\n if pr ntflg then\n print(``);\n print(`Linear DE . . `,c3*Diff(yx, x$2)+c2*Diff(yx,x)+c1*yx=c0);\n end if;\n\n if x0<>0 then\n c 0 := eval(subs(x=x+x0,c0));\n if prntflg then\n print(``) ;\n print(`Change the independent variable to `*x=x-x0,` so th at `*y(0)=y0);\n print(`and solve . . `,c3*Diff(yx,x$2)+c2*Dif f(yx,x)+c1*yx=c0);\n end if;\n end if;\n\n Gs := traperror(in ttrans[laplace](c0,x,ss));\n if Gs=lasterror or indets(Gs,'specfunc( anything,laplace)')<>\{\} then\n error \"could not determine the \+ Laplace transform of %1\",c0;\n end if;\n\n if prntflg then\n \+ print(``);\n print(`Taking Laplace transforms gives . .`);\n \+ print(c3*ss^2*L*[yx]-c3*ss*y0-c3*y1+c2*ss*L*[yx]-c2*y0+c1*L*[yx] = G s);\n print(``);\n print(`so . . `*L*[yx]*(c3*ss^2+c2*ss+c1) = Gs+c3*ss*y0+c3*y1+c2*y0);\n end if;\n\n Hs := normal((Gs+c3*ss *y0+c3*y1+c2*y0)/(c3*ss^2+c2*ss+c1)); \n \n if prntflg then\n \+ print(``);\n print(`Then . . `*L*[yx] = Hs);\n end if;\n\n \+ if has(indets(Hs,specfunc(anything,'exp')),ss) then\n Ps := map(s implify,frontend(expand,[Hs]));\n if Ps<>Hs then\n Hs := \+ Ps;\n if prntflg then\n print(``);\n pri nt(``= Hs);\n end if;\n end if; \n if type(Hs,`+`) t hen\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('anythin g','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,pa rfrac,ss);\n Ps := Ps + Ks: \n Ts := front end(expand,[Ks]);\n else\n Ps := Ps + Ts:\n \+ end if;\n Tx := traperror(inttrans[invlaplace](Ts ,ss,x));\n if Tx=lasterror or indets(Tx,'specfunc(anything, invlaplace)')<>\{\} then\n error \"could not determine t he inverse Laplace transform of %1\",Ts;\n end if;\n \+ Yx := Yx + Tx;\n end 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)],'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 := Hs/as;\n if type (Ns,ratpoly('anything',ss)) then \n Ts := frontend(expand,[ as*convert(Ns,parfrac,ss)]);\n if Ts<>Hs then\n \+ Hs := Ts;\n if prntflg then\n print( ``);\n print(``= Hs);\n end if;\n \+ end if; \n end if;\n Yx := traperror(inttrans[in vlaplace](Hs,ss,x));\n if Yx=lasterror or indets(Yx,'specfunc( anything,invlaplace)')<>\{\} then\n error \"could not deter mine the inverse Laplace transform of %1\",Hs;\n end if;\n \+ end if;\n else\n if type(Hs,ratpoly('anything',ss)) then \n \+ Ts := convert(Hs,parfrac,ss);\n if Ts<>Hs then\n \+ Hs := Ts;\n if prntflg then\n print(``); \n print(``= Hs);\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 prnt flg then\n print(``);\n print(``= Hs);\n \+ end if;\n end if;\n end if;\n Yx := traper ror(inttrans[invlaplace](Hs,ss,x));\n if Yx=lasterror or indets(Y x,'specfunc(anything,invlaplace)')<>\{\} then\n error \"could \+ not determine the inverse Laplace transform of %1\",Hs;\n end if; \n end if;\n\n if has(Yx,\{sinh,cosh\}) then Yx := simplify(conver t(Yx,exp),exp) end if;\n\n if x0<>0 then\n if prntflg then\n \+ print(``);\n print(`In terms of the shifted independent \+ variable . . `);\n print(yx=Yx);\n end if;\n Yx := e val(subs(x=x-x0,Yx));\n if prntflg then\n print(``);\n \+ print(`Returning to original independent variable`,x=x+x0);\n \+ print(yx=Yx);\n end if;\n end if;\n if prntflg then pri nt(``) end if;\n\n return yx=Yx;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {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 follo wing sections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolveL2" }{TEXT -1 11 ": examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 " Example 1" }}{PARA 256 "" 0 "" {TEXT -1 6 " 2 " }{XPPEDIT 18 0 "d^2 *y/(d*t^2)+dy/dt-y = t^2+3;" "6#/,(*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"t GF'F)!\"\"F)*&%#dyGF)%#dtGF-F)F(F-,&*$F,F'F)\"\"$F)" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "y(0) = 2 " "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`y '`(0) = 3;" "6#/-%$y~'G6#\"\"!\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "de := 2*diff(y(t),t$2)+diff(y(t),t)-y(t)=t^2+3;\nic \+ := y(0)=2,D(y)(0)=3;\ndesolveL2(\{de,ic\},y(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&\"\"#\"\"\"-%%diffG6$-%\"yG6#%\"t G-%\"$G6$F0F(F)F)-F+6$F-F0F)F-!\"\",&*$)F0F(F)F)\"\"$F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"#/--%\"DG6#F(F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Linear~DE~.~.~G/,(*&\"\"#\"\"\"-%%DiffG6$-%\"yG6#%\"tG-%\"$G6$F/F 'F(F(-F*6$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#/,,**\"\"#\" \"\")%\"sGF&F'%\"LGF'7#-%\"yG6#%\"tGF'F'*&\"\"%F'F)F'!\"\"\"\")F2*(F)F 'F*F'F+F'F'*&F*F'F+F'F2,&*&F&F'F)!\"$F'*&\"\"$F'F)F2F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%(so~.~. ~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,(*&\"\"#F&)%\"sGF/F&F&F1F&F&!\"\"F&,* *&F/F&F1!\"$F&*&\"\"$F&F1F2F&*&\"\"%F&F1F&F&\"\")F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~.~.~ G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&*(,*\"\"#F&*&\"\"$F&)%\"sGF/F&F&*&\"\"% F&)F3F5F&F&*&\"\")F&)F3F1F&F&F&F3!\"$,(*&F/F&F2F&F&F3F&F&!\"\"F=" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G,,*&\"\"\"F'*&\"\"$F',&%\"sGF'F'F'F'!\"\"F'*(\"#kF'F)F,,&*&\"\"#F 'F+F'F'F'F,F,F'*&F1F'F+!\"#F,*&\"\"*F'F+F,F,*&F1F'F+!\"$F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" yG6#%\"tG,,*&#\"\"\"\"\"$F+-%$expG6#,$F'!\"\"F+F+*&#\"#KF,F+-F.6#,$*& \"\"#F1F'F+F+F+F+*&F9F+F'F+F1\"\"*F1*$)F'F9F+F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de, ic\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,,*&#\"\" \"\"\"$F+-%$expG6#,$F'!\"\"F+F+*&#\"#KF,F+-F.6#,$*&\"\"#F1F'F+F+F+F+*& F9F+F'F+F1\"\"*F1*$)F'F9F+F1" }}}{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 6 " " } {XPPEDIT 18 0 "d^2*y/(d*t^2)+2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$% \"tGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt+4*y = 6*exp (-t);" "6#/,&*&%#dyG\"\"\"%#dtG!\"\"F'*&\"\"%F'%\"yGF'F'*&\"\"'F'-%$ex pG6#,$%\"tGF)F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"y G6#\"\"!F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 1;" "6#/-%$y~ 'G6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "de := diff(y(t),t$2)+2*diff (y(t),t)+4*y(t)=6*exp(-t);\nic := y(0)=0,D(y)(0)=1;\ndesolveL2(\{de,ic \},y(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%dif fG6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&\"\"%F2F*F2 F2,$*&\"\"'F2-%$expG6#,$F-!\"\"F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Linear~DE~.~.~ G/,(-%%DiffG6$-%\"yG6#%\"tG-%\"$G6$F,\"\"#\"\"\"*&F0F1-F'6$F)F,F1F1*& \"\"%F1F)F1F1,$*&\"\"'F1-%$expG6#,$F,!\"\"F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Laplace ~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**()%\"sG \"\"#\"\"\"%\"LGF)7#-%\"yG6#%\"tGF)F)F)!\"\"**F(F)F'F)F*F)F+F)F)*(\"\" %F)F*F)F+F)F),$*&\"\"'F),&F'F)F)F)F0F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\"\"\"%\"LGF &7#-%\"yG6#%\"tGF&,(*$)%\"sG\"\"#F&F&*&F1F&F0F&F&\"\"%F&F&,&*&\"\"'F&, &F0F&F&F&!\"\"F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF &*(,&\"\"(F&%\"sGF&F&,&F0F&F&F&!\"\",(*$)F0\"\"#F&F&*&F6F&F0F&F&\"\"%F &F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&,&*&\"\"#\"\"\"%\"sGF*!\"\"F*F,F*,(*$)F+F)F*F** &F)F*F+F*F*\"\"%F*F,F**&F)F*,&F+F*F*F*F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,( *(\"\"#\"\"\"-%$expG6#,$F'!\"\"F+-%$cosG6#*&\"\"$#F+F*F'F+F+F0*&#F+F5F +*(F5F6F,F+-%$sinGF3F+F+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,(*(\"\"#\"\"\"-%$ex pG6#,$F'!\"\"F+-%$cosG6#*&\"\"$#F+F*F'F+F+F0*&#F+F5F+*(F5F6F,F+-%$sinG F3F+F+F+*&F*F+F,F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "d^2* y/(d*t^2)-4*y = 4*sin(2*t)-4*cos(2*t);" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"* &F&F)*$%\"tGF'F)!\"\"F)*&\"\"%F)F(F)F-,&*&F/F)-%$sinG6#*&F'F)F,F)F)F)* &F/F)-%$cosG6#*&F'F)F,F)F)F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "de := diff(y(t),t$ 2)-4*y(t)=4*sin(2*t)-4*cos(2*t);\ndesolveL2(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\" \"#\"\"\"*&\"\"%F2F*F2!\"\",&*&F4F2-%$sinG6#,$*&F1F2F-F2F2F2F2*&F4F2-% $cosGF:F2F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%/Linear~DE~.~.~G/,&-%%DiffG6$-%\"yG6#%\"tG-%\"$G6$F, \"\"#\"\"\"*&\"\"%F1F)F1!\"\",&*&F3F1-%$sinG6#,$*&F0F1F,F1F1F1F1*&F3F1 -%$cosGF9F1F4" }}{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-6#\"\"!F)!\"\"--%\"DG6#F-F2F4*(F'F)F*F)7#F3F)F)*(\"\"%F)F*F)F +F)F4,&*&\"\")F),&*$F&F)F)F " 0 "" {MPLTEXT 1 0 38 "dsolve(\{de,y(0)=y0,D(y)(0)= Dy0\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,**&-%$e xpG6#,$*&\"\"#\"\"\"F'F0F0F0,&*&\"\"%!\"\"%$Dy0GF0F0*&F/F4%#y0GF0F0F0F 0*&-F+6#,$*&F/F0F'F0F4F0,(*&F3F4F5F0F4*&F/F4F7F0F0#F0F/F4F0F0*&#F0F/F0 -%$cosGF,F0F0*&#F0F/F0-%$sinGF,F0F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "d^2*y/(d*t^2)-5;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$% \"tGF&F(!\"\"F(\"\"&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt+4*y = 8* exp(t);" "6#/,&*&%#dyG\"\"\"%#dtG!\"\"F'*&\"\"%F'%\"yGF'F'*&\"\")F'-%$ expG6#%\"tGF'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 0" "6#/-%\"yG6 #\"\"\"\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 0;" "6#/-%$ y~'G6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "de := diff(y(t),t$2)-5*diff (y(t),x)+4*y(t)=8*exp(t);\nic := y(1)=0,D(y)(1)=0;\ndesolveL2(\{de,ic \},y(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%dif fG6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2F*F2F2,$*&\"\")F2-%$exp GF,F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"\"\"\" !/--%\"DG6#F(F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%/Linear~DE~.~.~G/,&-%%DiffG6$-%\"yG6#%\"tG-%\"$ G6$F,\"\"#\"\"\"*&\"\"%F1F)F1F1,$*&\"\")F1-%$expGF+F1F1" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/*&%DChan ge~the~independent~variable~to~G\"\"\"%\"tGF&,&F'F&F&!\"\"/*&%*~so~tha t~GF&-%\"yG6#\"\"!F&F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/and~solve~ .~.~G/,&-%%DiffG6$-%\"yG6#%\"tG-%\"$G6$F,\"\"#\"\"\"*&\"\"%F1F)F1F1,$* &\"\")F1-%$expG6#,&F,F1F1F1F1F1" }}{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)7#\"\"!F)F)*(\"\"%F)F*F)F+F)F),$*(\"\")F)- %$expG6#F)F),&F'F)F)!\"\"F " 0 "" {MPLTEXT 1 0 33 "dsolve(\{de,ic\},y(t) ):\ncombine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,(*&# \"\"%\"\"&\"\"\"*&-%$expG6#F-F--%$sinG6#,&*&\"\"#F-F'F-F-F7!\"\"F-F-F8 *&#\"\")F,F-*&F/F--%$cosGF4F-F-F8*&#F;F,F--F0F&F-F-" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*t^2)+4*y = 4*sin(2*t)-4*cos( 2*t);" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)*&\"\"%F)F (F)F),&*&F/F)-%$sinG6#*&F'F)F,F)F)F)*&F/F)-%$cosG6#*&F'F)F,F)F)F-" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 4" "6#/-%\"yG6#\"\"!\"\"%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 0;" "6#/-%$y~'G6#\"\"!F'" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 109 "de := diff(y(t),t$2)+4*y(t)=4*sin(2*t)-4*cos( 2*t);\nic := y(0)=4,D(y)(0)=0;\ndesolveL2(\{de,ic\},y(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tG-%\" $G6$F-\"\"#\"\"\"*&\"\"%F2F*F2F2,&*&F4F2-%$sinG6#,$*&F1F2F-F2F2F2F2*&F 4F2-%$cosGF9F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG 6#\"\"!\"\"%/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Linear~DE~.~.~G/,&-%%DiffG6$-%\"yG6 #%\"tG-%\"$G6$F,\"\"#\"\"\"*&\"\"%F1F)F1F1,&*&F3F1-%$sinG6#,$*&F0F1F,F 1F1F1F1*&F3F1-%$cosGF8F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Laplace~transforms~gives~.~ .G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**()%\"sG\"\"#\"\"\"%\"LGF)7#- %\"yG6#%\"tGF)F)*&\"\"%F)F'F)!\"\"*(F'F)F*F)7#\"\"!F)F)*(F1F)F*F)F+F)F ),&*&\"\")F),&*$F&F)F)F1F)F2F)*(F1F)F'F)F:F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\" \"\"%\"LGF&7#-%\"yG6#%\"tGF&,&*$)%\"sG\"\"#F&F&\"\"%F&F&,(*&\"\")F&F-! \"\"F&*(F2F&F0F&F-F6F6*&F2F&F0F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~.~.~G\"\"\"%\"LGF&7#- %\"yG6#%\"tGF&,$*(\"\"%F&,(\"\"#F&*&\"\"$F&%\"sGF&F&*$)F4F3F&F&F&,&*$) F4F1F&F&F/F&!\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&,&\"\")\"\"\"*&\"\"%F)%\"sGF)!\"\"F),&* $)F,\"\"#F)F)F+F)!\"#F)*(F+F)F,F)F.F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,&*&,&F'!\" \"\"\"%\"\"\"F--%$cosG6#,$*&\"\"#F-F'F-F-F-F-*&,&#F-F3F-F'F+F--%$sinGF 0F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "desolve(\{de,ic\},y(t)):\ncombine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,**&#\"\"\"\"\"#F+-%$sinG6#,$*&F,F+F 'F+F+F+F+*&F'F+-%$cosGF/F+!\"\"*&F'F+F-F+F5*&\"\"%F+F3F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 256 "" 0 " " {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*t^2)-y = t*sin(t)-cos(t); " "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)F(F-,&*&F,F)-%$ sinG6#F,F)F)-%$cosG6#F,F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0; " "6#/-%\"yG6#\"\"!F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 0; " "6#/-%$y~'G6#\"\"!F'" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "de := diff(y(t),t$2) -y(t)=t*sin(t)-cos(t);\nic := y(0)=0,D(y)(0)=0;\ndesolveL2(\{de,ic\},y (t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$ -%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"\"F*!\"\",&*&F-F2-%$sinGF,F2F2-%$cosGF ,F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"D G6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Linear~DE~.~.~G/,&-%%DiffG6$-%\"yG6#%\"tG-%\"$G6$F,\" \"#\"\"\"F)!\"\",&*&F,F1-%$sinGF+F1F1-%$cosGF+F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Laplace ~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*()%\"sG \"\"#\"\"\"%\"LGF)7#-%\"yG6#%\"tGF)F)*(F'F)F*F)7#\"\"!F)F)*&F*F)F+F)! \"\",&*(F(F)F'F),&*$F&F)F)F)F)!\"#F)*&F'F)F7F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\" \"\"%\"LGF&7#-%\"yG6#%\"tGF&,&*$)%\"sG\"\"#F&F&F&!\"\"F&,&*(F1F&F0F&,& F.F&F&F&!\"#F&*&F0F&F5F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~.~.~G\"\"\"%\"LGF&7#-%\"yG6# %\"tGF&,$*&%\"sGF&,&*$)F/\"\"#F&F&F&F&!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,$ *&#\"\"\"\"\"#F+*&F'F+-%$sinGF&F+F+!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "dsolve(\{de,ic\},y(t) ):\ncombine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,$*&# \"\"\"\"\"#F+*&F'F+-%$sinGF&F+F+!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 7" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "d^2*y/(d*t^2)-2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$% \"tGF&F(!\"\"F(F&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt+y = exp(t); " "6#/,&*&%#dyG\"\"\"%#dtG!\"\"F'%\"yGF'-%$expG6#%\"tG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "`y '`(0) = 2;" "6#/-%$y~'G6#\"\"!\"\"#" }{TEXT -1 3 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "de := diff(y(t),t$2)-2*diff(y(t),t)+y(t)=exp(t);\nic := y(0)=0,D(y)(0)=2;\ndesolveL2(\{de,ic\},y(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\" \"#\"\"\"*&F1F2-F(6$F*F-F2!\"\"F*F2-%$expGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Lin ear~DE~.~.~G/,(-%%DiffG6$-%\"yG6#%\"tG-%\"$G6$F,\"\"#\"\"\"*&F0F1-F'6$ F)F,F1!\"\"F)F1-%$expGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Laplace~transforms~gives~.~.G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**()%\"sG\"\"#\"\"\"%\"LGF)7#-% \"yG6#%\"tGF)F)F(!\"\"**F(F)F'F)F*F)F+F)F0*&F*F)F+F)F)*&F)F),&F'F)F)F0 F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,(*$)%\"sG\" \"#F&F&*&F1F&F0F&!\"\"F&F&F&,&*&F&F&,&F0F&F&F3F3F&F1F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~. ~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&*(,&*&\"\"#F&%\"sGF&F&F&!\"\"F&,&F1F &F&F2F2,(*$)F1F0F&F&*&F0F&F1F&F2F&F&F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"\"F'*$),&%\"s GF'F'!\"\"\"\"$F'F,F'*&\"\"#F'F*!\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,$*&#\"\"\" \"\"#F+*&,&*$)F'F,F+F+*&\"\"%F+F'F+F+F+-%$expGF&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+*&)F'F,F+-%$expGF&F+F+F+*(F,F+F'F+F/F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*t^2)+y = sqrt(t);" "6#/,&*(% \"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)F(F)-%%sqrtG6#F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 0;" "6#/-%$y~'G6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "de := diff(y(t),t$2)+y(t)=sqrt(t);\nic := y(0)=0,D(y )(0)=0;\ndesolveL2(\{de,ic\},y(t),info=true);\ng := unapply(rhs(%),t): " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tG-% \"$G6$F-\"\"#\"\"\"F*F2*$F-#F2F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Linear~DE~.~.~G/,&-%%Di ffG6$-%\"yG6#%\"tG-%\"$G6$F,\"\"#\"\"\"F)F1*$F,#F1F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Lapl ace~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*()%\" sG\"\"#\"\"\"%\"LGF)7#-%\"yG6#%\"tGF)F)*(F'F)F*F)7#\"\"!F)F)*&F*F)F+F) F),$*(F(!\"\"%#PiG#F)F(F'#!\"$F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\"\"\"%\"LGF&7#-% \"yG6#%\"tGF&,&*$)%\"sG\"\"#F&F&F&F&F&,$*(F1!\"\"%#PiG#F&F1F0#!\"$F1F& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,$**\"\"#!\"\"%#PiG #F&F/%\"sG#!\"$F/,&*$)F3F/F&F&F&F&F0F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,$*&#\"\"% \"#:\"\"\"*&)F'#\"\"&\"\"#F--%*hypergeomG6%7#F-7$#\"\"(F+#\"\"*F+,$*&F +!\"\"F'F2F>F-F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> 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+=;)*)H#F_\\l$\"+)oqJm&FI7$$\"+#4t^I#F_\\l$\"+RzMccFI7$$\"+nXO6BF_\\l$ \"+l\"[ik&FI7$$\"+UgbKUBF_\\l$\"+^iB\\bFI7$$\"+cvsoBF_\\l$\"+v)>DT&FI7$$\"+qJ8&R#F_ \\l$\"+^M)zB&FI7$$\"+b-oXCF_\\l$\"+_Ri][FI7$$\"#DF)$\"+)>$yjWFI-%'COLO URG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"tG%%y(t)G-%*THICKNESSG6#\" \"#-%%VIEWG6$;F(Fidm%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "de := diff(y(t),t$2)+y(t)=s qrt(t);\nic := y(0)=0,D(y)(0)=0;\ndsolve(\{de,ic\},y(t),method='laplac e');\ng := unapply(rhs(%),t):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#de G/,&-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"\"F*F2*$F-#F2F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F) F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,$*&#\"\"%\"#:\"\" \"*&)F'#\"\"&\"\"#F--%*hypergeomG6%7#F-7$#\"\"(F+#\"\"*F+,$*&F+!\"\"F' F2F>F-F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 9 " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*t^2)+d y/dt+101/4;" "6#,(*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&%#d yGF(%#dtGF,F(*&\"$,\"F(\"\"%F,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "y = \+ 1-H(t-1);" "6#/%\"yG,&\"\"\"F&-%\"HG6#,&%\"tGF&F&!\"\"F," }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`y '`(0) = 0;" "6#/-%$y~'G6#\"\"!F'" }{TEXT -1 2 ", \+ " }}{PARA 257 "" 0 "" {TEXT -1 40 "where H is the Heaviside step funct ion. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "u := t -> 1-Heaviside(t-1):\n'u(t)'=u(t);\nplot(u(t), t=0..4,color=red,thickness=2,ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6#%\"tG,&\"\"\"F)-%\"HG6#,&F'F)F)!\"\"F." }} {PARA 13 "" 1 "" {GLPLOT2D 510 173 173 {PLOTDATA 2 "6(-%'CURVESG6#7hn7 $$\"\"!F)$\"\"\"F)7$$\"3Hmmmm;')=()!#>F*7$$\"3RLLLe'40j\"!#=F*7$$\"3mm mm;6m$[#F3F*7$$\"3fmmm;yYULF3F*7$$\"3%HLL$eF>(>%F3F*7$$\"3Qmmm\">K'*) \\F3F*7$$\"3P*****\\Kd,\"eF3F*7$$\"3-mmm\"fX(emF3F*7$$\"3.*****\\U7Y]( F3F*7$$\"3'QLLLV!pu$)F3F*7$$\"3K+++DI(yv)F3F*7$$\"3xmmm;c0T\"*F3F*7$$ \"3))****\\P?uc$*F3F*7$$\"3+LLLe%GCd*F3F*7$$\"37++vo;F!o*F3F*7$$\"37mm ;z[6)y*F3F*7$$\"3d)*\\P%[O?%)*F3F*7$$\"37KLe*3ef*)*F3F*7$$\"3!*)\\(=#* )=H#**F3F*7$$\"3ol;z%pz)\\**F3F*7$$\"3YKeR(\\So(**F3F*7$$\"3#*******H, Q+5!#FjoF(7$$ \"3immmTc-)*>FjoF(7$$\"3Mmm;f`@'3#FjoF(7$$\"3y****\\nZ)H;#FjoF(7$$\"3Y mmmJy*eC#FjoF(7$$\"3')******R^bJBFjoF(7$$\"3f*****\\5a`T#FjoF(7$$\"3o* ***\\7RV'\\#FjoF(7$$\"3k*****\\@fke#FjoF(7$$\"3/LLL`4NnEFjoF(7$$\"3#** *****\\,s`FFjoF(7$$\"3[mm;zM)>$GFjoF(7$$\"3$*******pfa " 0 "" {MPLTEXT 1 0 181 "alias(H=Hea viside):\nu := t -> 1-Heaviside(t-1):\nde := diff(y(t),t$2)+diff(y(t), t)+101/4*y(t)=u(t);\nic := y(0)=0,D(y)(0)=0;\ndesolveL2(\{de,ic\},y(t) ,info=true);\nv := unapply(rhs(%),t):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&# \"$,\"\"\"%F2F*F2F2,&F2F2-%\"HG6#,&F-F2F2!\"\"F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Linear~ DE~.~.~G/,(*&\"\"%\"\"\"-%%DiffG6$-%\"yG6#%\"tG-%\"$G6$F/\"\"#F(F(*&F' F(-F*6$F,F/F(F(*&\"$,\"F(F,F(F(,&F'F(*&F'F(-%\"HG6#,&F/F(F(!\"\"F(F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Laplace~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(**\"\"%\"\"\")%\"sG\"\"#F'%\"LGF'7#-%\"yG6#%\"tGF'F' **F&F'F)F'F+F'F,F'F'*(\"$,\"F'F+F'F,F'F',&*&F&F'F)!\"\"F'*(F&F'-%$expG 6#,$F)F6F'F)F6F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,(*& \"\"%F&)%\"sG\"\"#F&F&*&F/F&F1F&F&\"$,\"F&F&,&*&F/F&F1!\"\"F&*(F/F&-%$ expG6#,$F1F7F&F1F7F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF &,$**\"\"%F&,&F&!\"\"-%$expG6#,$%\"sGF1F&F&F6F1,(*&F/F&)F6\"\"#F&F&*&F /F&F6F&F&\"$,\"F&F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(\"\"%\"\"\"%\"sG!\"\",(*&F'F()F)\"\" #F(F(*&F'F(F)F(F(\"$,\"F(F*F(**F'F(-%$expG6#,$F)F*F(F)F*F+F*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G,**(\"\"%\"\"\"\"$,\"!\"\"%\"sGF*F(**\"#;F(F)F*,&F+F(F(F(F(,(*&F' F()F+\"\"#F(F(*&F'F(F+F(F(F)F(F*F**&#F'F)F(*&-%$expG6#,$F+F*F(F+F*F(F* *&#F-F)F(*(F7F(F.F(F/F*F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,**&#\"\"%\"$,\"\"\"\"*& -%$expG6#,$*&\"\"#!\"\"F'F-F5F--%$cosG6#,$*&\"\"&F-F'F-F-F-F-F5*&#F4\" $0&F-*&F/F--%$sinGF8F-F-F5#F+F,F-*&,(*&FBF-*&-F06#,&*&F4F5F'F-F5#F-F4F -F--F76#,&*&F;F-F'F-F-F;F5F-F-F-*&#F4F>F-*&FGF--FAFMF-F-F-#F+F,F5F--% \"HG6#,&F'F-F-F5F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(v(t),t=0..5,color=green,thickness=2) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 541 243 243 {PLOTDATA 2 "6'-%'CURVESG6 #7\\s7$$\"\"!F)F(7$$\"3gmmTN@Ki8!#>$\"3@FU$o93SB*!#A7$$\"3ALL$3FWYs#F- $\"3Wld^r#GEn$!#@7$$\"3%)***\\iSmp3%F-$\"3IPt_8j=5#)F67$$\"3WmmmT&)G\\ aF-$\"3c9t[$ew!\\9!#?7$$\"3m****\\7G$R<)F-$\"3xlJ&zthi?$FA7$$\"3GLLL3x &)*3\"!#=$\"3C-Bc\\DA)e&FA7$$\"3))**\\i!R(*Rc\"FJ$\"3%HG\")*oxt.6F-7$$ \"3umm\"H2P\"Q?FJ$\"3%\\IIHXV5y\"F-7$$\"3!***\\PMnNrDFJ$\"3%R*Q]HlkYEF -7$$\"3MLL$eRwX5$FJ$\"3'p%G\"R^+'eNF-7$$\"3rLLL$eI8k$FJ$\"3%))G,5bMoX% F-7$$\"33ML$3x%3yTFJ$\"31<)*>_r68a6I_oF-7$$\"3#pmm;HZ_O'FJ$\"3'>ZYwpF1&oF-7$$\" 3I++vVVX$\\'FJ$\"3+%>/@wgq$oF-7$$\"3pLL$eRh;i'FJ$\"3!))Q@ey2=\"oF-7$$ \"31nm\"zWo)\\nFJ$\"3#)*Q[&R<7vnF-7$$\"3%QL$3_DG1qFJ$\"3O5G`%3!oomF-7$ $\"3]***\\il'pisFJ$\"3oSG84Yg?lF-7$$\"3Qnm\"HKkIz(FJ$\"3A<&*HKxy(4'F-7 $$\"3>MLe*)>VB$)FJ$\"3Of*f=?I)\\bF-7$$\"3wmmTg()4_))FJ$\"3Zpv=1P(H#\\F -7$$\"3Y++DJbw!Q*FJ$\"3?#R#z-*=EE%F-7$$\"3+N$ekGkX#**FJ$\"3qq2+;*Qxf$F -7$$\"3%ommTIOo/\"!#<$\"35!z(>B%[n)GF-7$$\"3E+]7GTt%4\"Fgu$\"3>qy\\Ga. >@F-7$$\"3YLL3_>jU6Fgu$\"3GLo'fhXzE\"F-7$$\"3C+voaFfp6Fgu$\"3.u$pu?#o; 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" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "alias(H=Heaviside):\nh \+ := t -> t+H(t-1)*(1-t)+H(t-2)*(2-t)+H(t-4)*(t-4)+H(t-5)*(t-5)+H(t-6)*( 6-t):\nde := diff(y(t),t$2)+diff(y(t),t)+25/2*y(t)=h(t);\nic := y(0)=0 ,D(y)(0)=0;\ndesolveL2(\{de,ic\},y(t),info=true);\ng := unapply(rhs(%) ,t):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\" tG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&#\"#DF1F2F*F2F2,.F-F2*&-%\"HG6#,&F 2!\"\"F-F2F2,&F2F2F-F>F2F2*&-F;6#,&F-F2F1F>F2,&F1F2F-F>F2F2*&-F;6#,&F- F2\"\"%F>F2FHF2F2*&-F;6#,&F-F2\"\"&F>F2FMF2F2*&-F;6#,&F-F2\"\"'F>F2,&F SF2F-F>F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F */--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 " " 1 "" {XPPMATH 20 "6$%/Linear~DE~.~.~G/,(*&\"\"#\"\"\"-%%DiffG6$-%\"y G6#%\"tG-%\"$G6$F/F'F(F(*&F'F(-F*6$F,F/F(F(*&\"#DF(F,F(F(,8*&F'F(F/F(F (*&F'F(-%\"HG6#,&F(!\"\"F/F(F(F(*(F'F(F/F(F;F(F?*&\"\"%F(-F<6#,&F/F(F' F?F(F(*(F'F(F/F(FCF(F?*(F'F(-F<6#,&F/F(FBF?F(F/F(F(*&\"\")F(FHF(F?*(F' F(-F<6#,&F/F(\"\"&F?F(F/F(F(*&\"#5F(FNF(F?*&\"#7F(-F<6#,&F/F(\"\"'F?F( F(*(F'F(FVF(F/F(F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Laplace~transforms~gives~.~.G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(**\"\"#\"\"\")%\"sGF&F'%\"LGF'7#-% \"yG6#%\"tGF'F'**F&F'F)F'F*F'F+F'F'*(\"#DF'F*F'F+F'F',.*&F&F'F)!\"#F'* (F&F'-%$expG6#,$F)!\"\"F'F)F5F;*(F&F'-F86#,$*&F&F'F)F'F;F'F)F5F;*(F&F' -F86#,$*&\"\"%F'F)F'F;F'F)F5F'*(F&F'-F86#,$*&\"\"&F'F)F'F;F'F)F5F'*(F& F'-F86#,$*&\"\"'F'F)F'F;F'F)F5F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! 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=0,D(y)(0)=0;\ndesolveL2(\{de,ic\},y(t),info=true);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"\" *(F1F2%'lambdaGF2-F(6$F*F-F2F2*&)%&omegaGF1F2F*F2F2-%$cosG6#*&F9F2F-F2 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6# F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Linear~DE~.~.~G/,(-%%DiffG6$-%\"yG6#%\"tG-%\"$G6$F,\" \"#\"\"\"*(F0F1%'lambdaGF1-F'6$F)F,F1F1*&)%&omegaGF0F1F)F1F1-%$cosG6#* &F8F1F,F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Laplace~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*()%\"sG\"\"#\"\"\"%\"LGF)7#-%\"yG6#%\"tGF)F)*, F(F)%'lambdaGF)F'F)F*F)F+F)F)*()%&omegaGF(F)F*F)F+F)F)*&F'F),&*$F&F)F) *$F3F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,(*$)%\"s G\"\"#F&F&*(F1F&%'lambdaGF&F0F&F&*$)%&omegaGF1F&F&F&*&F0F&,&F.F&F4F&! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&*(%\"sGF&, &*$)F.\"\"#F&F&*$)%&omegaGF2F&F&!\"\",(F0F&*(F2F&%'lambdaGF&F.F&F&F3F& F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"\"F'*(\"\"#F',&*$)%\"sGF)F'F'*$)%&omegaGF)F 'F'F'%'lambdaGF'!\"\"F'*&F'F'*(F)F',(F+F'*(F)F'F1F'F-F'F'F.F'F'F1F'F2F 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,&*&#\"\"\"\"\"%F(-%$expG6#*& ,&*$,&*$)%'lambdaG\"\"#F(F(*$)%&omegaGF4F(!\"\"#F(F4F(F3F8F(%\"tGF(F(F 8*&#F(F)F(-F+6#,$*&,&F/F(F3F(F(F:F(F8F(F(F(F3F8*&,&F3F(F7F8F(,&F3F(F7F (F(#F8F4F(*&F9F(*(F3F8F7F8-%$sinG6#*&F7F(F:F(F(F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG ,&*(,&*&#\"\"\"\"\"%F--%$expG6#*&,&*$,&*$)%'lambdaG\"\"#F-F-*$)%&omega GF9F-!\"\"#F-F9F-F8F=F-F'F-F-F=*&#F-F.F--F06#,$*&,&F4F-F8F-F-F'F-F=F-F -F-F8F=*&,&F8F-FF-*(F8F=F " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,(*&#\"\"\"\"\"%F+*(%'lambdaG!\"\",&*$)F. \"\"#F+F+*$)%&omegaGF3F+F/#F/F3-%$expG6#*&,&*$F0#F+F3F+F.F/F+F'F+F+F+F /*&#F+F,F+*(-F96#*&,&F.F/F=F/F+F'F+F+F.F/F0F7F+F+*&F>F+*(F.F/F6F/-%$si nG6#*&F6F+F'F+F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 14" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2/a ^2" "6#*&\"\"#\"\"\"*$%\"aGF$!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "d ^2*y/(d*t^2)+1/a;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F (*&F(F(%\"aGF,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt-y = t^2/a+3/2; " "6#/,&*&%#dyG\"\"\"%#dtG!\"\"F'%\"yGF),&*&%\"tG\"\"#%\"aGF)F'*&\"\"$ F'F.F)F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = b" "6#/-%\"yG6#\"\" !%\"bG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 0;" "6#/-%$y~'G6# \"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "de := 2/a^2*diff(y(t),t$2)+diff(y( t),t)/a-y(t)=t^2/a+3/2;\nic := y(0)=b,D(y)(0)=0;\ndesolveL2(\{de,ic\}, y(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*(\"\"#\" \"\"%\"aG!\"#-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F2F(F)F)*&-F-6$F/F2F)F*!\" \"F)F/F9,&*&F2F(F*F9F)#\"\"$F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #icG6$/-%\"yG6#\"\"!%\"bG/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Linear~DE~.~.~ G/,(*&\"\"%\"\"\"-%%DiffG6$-%\"yG6#%\"tG-%\"$G6$F/\"\"#F(F(*(F3F(%\"aG F(-F*6$F,F/F(F(*(F3F()F5F3F(F,F(!\"\",&*(F3F(F5F()F/F3F(F(*&\"\"$F(F9F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Laplace~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,**\"\"%\"\"\")%\"sG\"\"#F'%\"LGF'7#-%\"yG6#%\"t GF'F'*(F&F'F)F'%\"bGF'!\"\"*,F*F'%\"aGF'F)F'F+F'F,F'F'*(F*F'F5F'F2F'F3 **F*F')F5F*F'F+F'F,F'F3,&*(F&F'F5F'F)!\"$F'*(\"\"$F'F5F*F)F3F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/**%(so~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,(*&\"\"%F&)%\"sG\"\"#F&F& *(F2F&%\"aGF&F1F&F&*&F2F&)F4F2F&!\"\"F&,**(F/F&F4F&F1!\"$F&*(\"\"$F&F4 F2F1F7F&*(F/F&F1F&%\"bGF&F&*(F2F&F4F&F>F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~.~.~G \"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,$**\"\"#!\"\",**&\"\"%F&%\"aGF&F&*(\"\" $F&)F4F/F&)%\"sGF/F&F&*(F3F&)F9F3F&%\"bGF&F&**F/F&F4F&FF2F.F'F.F.F.F.*&)F'F/F .)F2F/F.F>\"\"'F>*(F/F.F2F.F'F.F>*(F3F.F/F>F2F3F>F.F2!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsol ve(\{de,ic\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG, (*&#\"\"\"\"\"'F+*(%\"aG!\"$,(\"\"%F+*&\"\"$F+)F.F3F+F+*(\"\"#F+F4F+% \"bGF+F+F+-%$expG6#,$*&F.F+F'F+!\"\"F+F+F+*&#F+F3F+*(F.F/,(\"#;F+*&F3F +F4F+F+*(F6F+F4F+F7F+F+F+-F96#,$*(F6F=F.F+F'F+F+F+F+F+*(F6F=,**(F6F+)F 'F6F+)F.F6F+F=\"#7F=*(F1F+F.F+F'F+F=*&F3F+F4F+F=F+F.F/F+" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 71 "S olve the following initial value problems by using Laplace transforms. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "d^2*y/(d*t^2)+7;" "6#,&*(%\"dG\"\"#%\"yG \"\"\"*&F%F(*$%\"tGF&F(!\"\"F(\"\"(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt+12*y = 3*t-2;" "6#/,&*&%#dyG\"\"\"%#dtG!\"\"F'*&\"#7F'%\"yGF'F' ,&*&\"\"$F'%\"tGF'F'\"\"#F)" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = \+ 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 0 ;" "6#/-%$y~'G6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "___________________________________________\n " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "___________________________________________ " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "d^2*y/(d*t^2)+dy/dt-6*y = 2*exp(-3*t);" "6#/,(*(%\"dG\"\"#%\"yG\"\"\" *&F&F)*$%\"tGF'F)!\"\"F)*&%#dyGF)%#dtGF-F)*&\"\"'F)F(F)F-*&F'F)-%$expG 6#,$*&\"\"$F)F,F)F-F)" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 1" "6# /-%\"yG6#\"\"!\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 0; " "6#/-%$y~'G6#\"\"!F'" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "___________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 43 "____________ _______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 " " {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*t^2)+5;" "6#,&*(%\"dG\"\"# %\"yG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(\"\"&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt+4*y = cos*2*t;" "6#/,&*&%#dyG\"\"\"%#dtG!\"\"F'*&\"\"%F'% \"yGF'F'*(%$cosGF'\"\"#F'%\"tGF'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y( 0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0 ) = 2;" "6#/-%$y~'G6#\"\"!\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "___________________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 43 "__ _________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }