{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 "" 264 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 264 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 267 "Times" 1 12 128 0 0 1 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 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 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 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 "Bulle t Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{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 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 258 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 46 "A procedure for solving 1st order linear DE's " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 10.10.2007" }} {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 "DE sol.m" }{TEXT -1 32 " is required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command simi lar 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 "Summary of method for solving 1st order linear differen tial equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 48 "In order to solve a linear differential e quation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+p(x) *y = q(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"pG6#%\"xGF'%\"yGF'F'- %\"qG6#F." }{TEXT -1 14 " ------- (i) " }}{PARA 0 "" 0 "" {TEXT -1 20 "first determine the " }{TEXT 263 18 "integrating factor" }{TEXT -1 1 " " }{XPPEDIT 18 0 "r(x) = exp(Int(p(x),x));" "6#/-%\"rG6#%\"xG-% $expG6#-%$IntG6$-%\"pG6#F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Multiplying (i) by " }{XPPEDIT 18 0 "u(x)" "6#-%\"uG6#%\"xG" } {TEXT -1 6 " gives" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "r(x);" "6#-%\"rG6#%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+r(x) *p(x)*y = r(x)*q(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(-%\"rG6#%\"xGF' -%\"pG6#F.F'%\"yGF'F'*&-F,6#F.F'-%\"qG6#F.F'" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "r(x)*y*`'`(x)+r(x)*p(x)*y(x) = r(x)*q(x);" "6#/,&*(-%\" rG6#%\"xG\"\"\"%\"yGF*-%\"'G6#F)F*F**(-F'6#F)F*-%\"pG6#F)F*-F+6#F)F*F* *&-F'6#F)F*-%\"qG6#F)F*" }{TEXT -1 14 " ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "r*`'`(x) = d/dx;" "6#/*&%\"rG\"\"\" -%\"'G6#%\"xGF&*&%\"dGF&%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[e xp(Int(p(x),x))];" "6#7#-%$expG6#-%$IntG6$-%\"pG6#%\"xGF-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(Int(p(x),x));" "6#/%!G-%$expG6#-%$IntG6$- %\"pG6#%\"xGF." }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\" %#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[Int(p(x),x)];" "6#7#-%$In tG6$-%\"pG6#%\"xGF*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = exp(Int(p(x),x))*p(x);" "6#/%!G*&-%$expG6#-%$In tG6$-%\"pG6#%\"xGF/\"\"\"-F-6#F/F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ r(x)*p(x);" "6#/%!G*&-%\"rG6#%\"xG\"\"\"-%\"pG6#F)F*" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 35 "so (ii) can be written in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r(x)*y*`'`(x)+r*`'` (x)*y(x) = r(x)*q(x);" "6#/,&*(-%\"rG6#%\"xG\"\"\"%\"yGF*-%\"'G6#F)F*F **(F'F*-F-6#F)F*-F+6#F)F*F**&-F'6#F)F*-%\"qG6#F)F*" }{TEXT -1 15 " --- ---- (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The left hand side of (iii) is the " }{TEXT 263 23 "derivative \+ of a product" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"% #dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[r(x)*y(x)];" "6#7#*&-%\"rG 6#%\"xG\"\"\"-%\"yG6#F(F)" }{TEXT -1 12 " so we have " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "d/dx;" "6#*&%\"dG \"\"\"%#dxG!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "[y(x)*r(x)] = r(x)* q(x);" "6#/7#*&-%\"yG6#%\"xG\"\"\"-%\"rG6#F)F**&-F,6#F)F*-%\"qG6#F)F* " }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y(x)*r(x) = Int(r(x)*q(x),x);" "6#/ *&-%\"yG6#%\"xG\"\"\"-%\"rG6#F(F)-%$IntG6$*&-F+6#F(F)-%\"qG6#F(F)F(" } {TEXT -1 2 ", " }}{PARA 258 "" 0 "" {TEXT -1 12 "which gives " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y(x) = Int(r(x)*q(x) ,x)/r(x);" "6#/-%\"yG6#%\"xG*&-%$IntG6$*&-%\"rG6#F'\"\"\"-%\"qG6#F'F0F 'F0-F.6#F'!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "r(x) = exp(Int (p(x),x));" "6#/-%\"rG6#%\"xG-%$expG6#-%$IntG6$-%\"pG6#F'F'" }{TEXT -1 3 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 266 26 "_________ _________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "A procedure for solving 1st order linear DE's: " }{TEXT 0 9 "desolveLN" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "desolveLN: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 18 "Calling Sequence:\n" }} {PARA 0 "" 0 "" {TEXT 260 2 " " }{TEXT -1 76 " desolveLN( de ) \n \+ desolveLN( \{de,ic\} )\n desolveLN( \{de,ic\},y(x) )" }{TEXT 261 1 "\n" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters: " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " \+ de - " }{TEXT -1 92 " a first order differential equation with the derivative given in the form diff(y(x),x)," }}{PARA 0 "" 0 "" {TEXT -1 94 " (if x and y are the independent a nd dependent variables respectively)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 23 10 " ic - " }{TEXT -1 50 " an initial condition in the form y(x0) = y0. " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }} {PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 9 "des olveLN" }{TEXT -1 62 " attempts to solve a first order linear differen tial equation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+p(x)*y = q(x);" "6#/,&*&%#dyG\"\" \"%#dxG!\"\"F'*&-%\"pG6#%\"xGF'%\"yGF'F'-%\"qG6#F." }{TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 33 "by using the integrating factor " } {XPPEDIT 18 0 "r(x) = exp(Int(p(x),x));" "6#/-%\"rG6#%\"xG-%$expG6#-%$ IntG6$-%\"pG6#F'F'" }{TEXT -1 9 ", so that" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y(x) = (Int(q(x)*r(x),x)+C[1])/r(x);" "6#/-% \"yG6#%\"xG*&,&-%$IntG6$*&-%\"qG6#F'\"\"\"-%\"rG6#F'F1F'F1&%\"CG6#F1F1 F1-F36#F'!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 90 "If no initial condition is given, a gener al solution with an arbitrary constant is sought." }}{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 the 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 the subsection, place the cursor anywhe re after the prompt [ > and press [Enter].\nYou can then close up the subsection. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "desolveLN: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9013 "desolveLN := proc() \n local ff,vars,derivs,x,yx,y,df,dff,drv,e1,e2,prntflg,Options,\n \+ j,sols,islog,opls,fx,gx,intg,intfact,val,soln,dy,dx,\n \+ initcond,de,ic,x0,y0,rsic,lsic,f0,v0,const,fact,\n factOK,intO K,factgx,intg2,startopts,xx,yy,ee,d,\n Cs,gt,jS,j1,s,i,nvars; \n global C;\n\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(equation)\}) an d 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 has(de,'D') then de := convert(de,'diff') \+ else de := ff end if;\n else\n error \"the 1st argument, %1, is invalid .. it should be an equation or a set (or list) of 2 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 \"2nd argument, %1, has incorrect form for the depende nt variable\",ee;\n end if;\n end if;\n end if;\n\n p rntflg := false;\n if nargs>=startopts then\n Options:=[args[st artopts..nargs]];\n if not type(Options,list(equation)) then\n \+ error \"each optional argument must be an equation\"\n end \+ if;\n if hasoption(Options,'info','prntflg','Options') 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 optio n for %2\",op(1,Options),procname;\n end if;\n end if;\n\n de rivs := indets(de,'specfunc(anything,diff)');\n if derivs=\{\} then \n error \"the 1st argument, %1, is invalid .. it should be a dif ferential equation or a set (or list) containing a differential equati on and an initial condition\",ff;\n end if;\n nvars := nops(indets (derivs,'name'));\n if nvars<>1 then\n if nvars=0 then\n \+ error \"there is a problem with the independent variable occurring i n the derivative(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 nvars<>1 then\n if nvars=0 then\n error \"there is a problem with the dependent variable occurring in the derivative(s)\" \n else\n error \"there should only be one dependent vari able in the differential equation\"\n end if;\n end if;\n\n i f nops(derivs)<>1 then\n error \"there are too many derivatives i n the differential equation .. note that the differential equation mus t be of order 1\"\n end if;\n df := op(1,derivs);\n if type(df,f unction) and op(0,df)=diff and nops(df)=2 then\n yx := op(1,df); \n if not type(yx,anyfunc(name)) then\n error \"the 1st a rgument %1, in the derivative, %2, is invalid .. it should be the 'unk nown' dependent variable\",yx,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 independent v ariable\",x,df;\n end if; \n else\n error \"the derivative , %1, does not make sense\",df;\n end if;\n\n y := op(0,yx);\n v ars := indets(de,name);\n if member(y,vars) then\n error \"%1 a nd %2 cannot both appear in the differential equation\",yx,y;\n end \+ if;\n if op(1,yx)<>x then\n error \"the derivative, %1, does no t make sense\",df;\n end if;\n\n if startopts = 3 then \n if \+ x<>xx or y<>yy then\n error \"cannot solve the ODE 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 \"initial condition must not involve %1 or %2\",x,y;\n \+ end if;\n else\n error \"initial condition is not deciph erable\"\n end if;\n rsic := rhs(ic);\n if type(rsic,al gebraic) then\n y0 := rsic;\n if has(y0,\{x,y\}) then \n error \"initial condition must not involve %1 or %2\",x, y;\n end if;\n else\n error \"initial condition i s not decipherable\"\n end if;\n end if;\n\n drv := solve(de, df);\n if nops([drv])=1 then\n e1 := subs(yx=y,factor(drv));\n \+ else\n error \"cannot obtain a unique expression for the deriva tive\"\n end if;\n\n if assigned(C) and not type(eval(C),table) th en\n C := table();\n WARNING(\"C has been redefined as a tab le for use as arbitrary constants\");\n end if;\n # Find the indic ex j1 to use in the constant\n Cs := select(type,indets(de),'specind ex(posint,C)');\n gt := proc(_u) local s,j;\n typematch(_ u,C[j::posint],'s'); \n subs(s,j)\n end proc:\n j S := sort([op(map(gt,Cs))]);\n for i to nops(jS)+1 do\n if not \+ member(i,jS) then j1 := i; break; end if; \n end do;\n \n # Ha ndle the special case first\n if not member(y,indets(e1,name)) then \n if prntflg then\n print(``);\n print(`Linear D E . . `,Diff(yx,x)=e1);\n print(``);\n end if;\n in tg := Int(e1,x);\n if prntflg then\n print(y=simplify(int g));print(``);\n end if;\n val := value(intg);\n intOK \+ := indets(val,'specfunc(anything,int)')=\{\};\n if prntflg then\n print(y=simplify(val+C[j1]));print(``);\n end if;\n \+ if initcond then\n if intOK then\n v0 := traperror (eval(subs(x=x0,val)));\n if v0=lasterror then\n \+ error \"impossible initial condition\"\n end if;\n \+ const := simplify(y0-v0);\n if prntflg then\n \+ print(`Applying the initial condition . . `,C[j1]=const);\n \+ print(``);\n end if;\n soln := yx=s implify(subs(C[j1]=const,val+C[j1]));\n else\n soln := yx=(Int(subs(x=_u,e1),_u=x0..x)+y0);\n end if;\n else \n if intOK then\n soln := yx=simplify(expand((val+ C[j1])));\n else\n soln := yx=Intat(subs(x=_u,e1),_ u=x)+C[j1];\n end if;\n end if;\n return soln;\n e nd if; # end of special case\n\n e2 := expand(e1);\n if not member (degree(e2,y),\{0,1\}) then\n error \"the DE is not linear\"\n \+ end if;\n fx := -traperror(simplify(coeff(e2,y,1)));\n if fx=laste rror then\n error \"the DE is not linear\"\n end if;\n\n gx : = combine(simplify(e2+fx*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(`Linear DE . . `,Diff(yx,x)+fx*yx =gx);\n print(``);\n end if;\n intg := Int(fx,x);\n if prnt flg then\n print(`Integrating factor . . `,exp(intg));\n end i f;\n val := value(intg);\n factOK := indets(val,'specfunc(anything ,int)')=\{\};\n if factOK then \n intfact := simplify(exp(val)) ;\n if prntflg then\n print(` = `,intfac t);print(``);\n end if;\n factgx := simplify(gx*intfact);\n \+ intg := Int(factgx,x);\n if prntflg then\n print(yx* intfact=intg);print(``);\n end if;\n val := value(intg);\n \+ intOK := indets(val,'specfunc(anything,int)')=\{\};\n if intO K then\n intg2 := val;\n else\n intg2 := Intat(su bs(x=_u,factgx),_u=x);\n end if; \n else\n intfact := exp(Intat(subs(x=_u,fx),_u=x));\n intg2 := Intat(subs(x=_v,gx )*\n exp(Intat(subs(x=_u,fx),_u=_v)),_v=x);\n end if;\n if prntflg then\n print(yx*intfact=intg2+C[j1]);print(``);\n \+ end if;\n \n if initcond then\n if indets(intfact,'specfunc( anything,int)')=\{\} then\n f0 := traperror(eval(subs(x=x0,int fact)));\n if f0=lasterror or f0=0 then\n error \"i mpossible initial condition\"\n end if;\n else\n \+ f0 := exp(Intat(fx,x=x0));\n end if;\n if indets(val,'specfu nc(anything,int)')=\{\} then\n v0 := traperror(eval(subs(x=x0, val)));\n if v0=lasterror then\n error \"impossible initial condition\"\n end if;\n const := simplify(y0* f0-v0);\n if prntflg then\n print(`Applying the ini tial condition . . `,C[j1]=const);\n print(``);\n \+ end if;\n fact := simplify(1/intfact);\n soln := yx=si mplify(fact*intg2)+const*fact;\n else\n const := y0*f0-In tat(subs(x=_v,intfact*gx),_v=x0);\n if prntflg then\n \+ print(`Applying the initial condition . . `,C[j1]=const);\n \+ print(``);\n end if;\n soln := yx=(Int(subs(x=_v, intfact*gx),_v=x0..x)+y0*f0)/intfact;\n end if;\n else\n f act := simplify(1/intfact);\n soln := yx=simplify(fact*intg2)+C[j 1]*fact;\n end if;\n return soln;\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 gi ven 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 "desolveLN" }{TEXT -1 27 ": general solution 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/dx+3*y = exp(2*x)" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*& \"\"$F'%\"yGF'F'-%$expG6#*&\"\"#F'%\"xGF'" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "de := diff(y(x),x)+3*y(x)=exp(2*x);\ndesolveLN(de,y(x),info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\" \"*&\"\"$F.F*F.F.-%$expG6#,$*&\"\"#F.F-F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~ ~G/,&-%%DiffG6$-%\"yG6#%\"xGF,\"\"\"*&\"\"$F-F)F-F--%$expG6#,$*&\"\"#F -F,F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$\"\"$%\"xG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G-%$expG6#,$ *&\"\"$\"\"\"%\"xGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$*&\"\"$\"\"\"%\"xGF+F+F+ -%\"yG6#F,F+-%$IntG6$-F&6#,$*&\"\"&F+F,F+F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$*& \"\"$\"\"\"%\"xGF+F+F+-%\"yG6#F,F+,&*&#F+\"\"&F+-F&6#,$*&F3F+F,F+F+F+F +&%\"CG6#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"&F+-%$expG6#,$*&\"\"#F+ F'F+F+F+F+*&&%\"CG6#F+F+-F.6#,$*&\"\"$F+F'F+!\"\"F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve( de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&-%$expG 6#,$F'\"\"&#\"\"\"F.%$_C1GF0F0-F+6#,$F'!\"$F0" }}}{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 "x;" "6#%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d y/dx-y = x^4;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF)*$%\"xG\"\"%" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "de := x*diff(y(x),x)-y(x)=x^4;\ndesolveLN(de,inf o=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&*&%\"xG\"\"\"-%%d iffG6$-%\"yG6#F(F(F)F)F-!\"\"*$)F(\"\"%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~ ~G/,&-%%DiffG6$-%\"yG6#%\"xGF,\"\"\"*&F,!\"\"F)F-F/*$)F,\"\"$F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$,$*&\"\"\"F,%\"xG!\"\"F. F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G*&\"\"\"F %%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&%\"xG!\"\"-%\"yG6#F%\"\"\"-%$IntG6$*$)F%\"\"#F*F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"xG!\"\"-%\"yG6#F%\"\"\",&*&\"\"$F&F%F-F*&%\"CG6#F*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG,&*&\"\"$!\"\"F'\"\"%\"\"\"*&&%\"CG6#F-F-F'F-F-" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG *&,&*$)F'\"\"$\"\"\"#F-F,%$_C1GF-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 "dy/dx+2*x*y = exp(-3*x)*(2*x-3);" "6#/,&*&%#dyG\"\" \"%#dxG!\"\"F'*(\"\"#F'%\"xGF'%\"yGF'F'*&-%$expG6#,$*&\"\"$F'F,F'F)F', &*&F+F'F,F'F'F4F)F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "de := diff(y(x),x)+2*x*y(x) =exp(-3*x)*(2*x-3);\ndesolveLN(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*(\"\"#F.F-F.F* F.F.*&-%$expG6#,$*&\"\"$F.F-F.!\"\"F.,&*&F0F.F-F.F.F7F8F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear ~DE~.~.~~G/,&-%%DiffG6$-%\"yG6#%\"xGF,\"\"\"*(\"\"#F-F,F-F)F-F-,&*(F/F --%$expG6#,$*&\"\"$F-F,F-!\"\"F-F,F-F-*&F7F-F2F-F8" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~ factor~.~.~~G-%$expG6#-%$IntG6$,$*&\"\"#\"\"\"%\"xGF-F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G-%$expG6#*$)%\"xG\"\"#\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#*$)%\"xG\"\"#\"\"\"F,-%\"yG6#F*F,-%$IntG6$* &-F&6#,&*&\"\"$F,F*F,!\"\"F(F,F,,&*&F+F,F*F,F,F8F9F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6 #*$)%\"xG\"\"#\"\"\"F,-%\"yG6#F*F,,&-F&6#,&*&\"\"$F,F*F,!\"\"F(F,F,&% \"CG6#F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&-%$expG6#,$*&\"\"$\"\"\"F'F/!\"\"F/*&&% \"CG6#F/F/-F*6#,$*$)F'\"\"#F/F0F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&-%$expG6#*&F'\"\"\",&!\"$F .F'F.F.F.%$_C1GF.F.-F+6#,$*$)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 4" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "dy/dx+tan(x)*y = sin(2*x);" "6#/,&*&%#dyG\"\" \"%#dxG!\"\"F'*&-%$tanG6#%\"xGF'%\"yGF'F'-%$sinG6#*&\"\"#F'F.F'" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "de := diff(y(x),x)+tan(x)*y(x)=sin(2*x);\ndesolv eLN(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diff G6$-%\"yG6#%\"xGF-\"\"\"*&-%$tanGF,F.F*F.F.-%$sinG6#,$*&\"\"#F.F-F.F. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$-%\"yG6#%\"xGF,\"\"\"*&-%$tanGF+F -F)F-F--%$sinG6#,$*&\"\"#F-F,F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~factor~.~.~~G-%$ex pG6#-%$IntG6$-%$tanG6#%\"xGF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~ ~~~~~~~~~~~~~=~~~G*&\"\"\"F%-%$cosG6#%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$cosG6#%\"x G!\"\"-%\"yGF'\"\"\"-%$IntG6$*&-%$sinG6#,$*&\"\"#F,F(F,F,F,F%F)F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*&-%$cosG6#%\"xG!\"\"-%\"yGF'\"\"\",&*&\"\"#F,F%F,F)&%\"CG6#F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG,&*&\"\"#\"\"\")-%$cosGF&F*F+!\"\"*&&%\"CG6#F+F+F-F+F+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"yG6#%\"xG,&*&\"\"#\"\"\")-%$cosGF&F*F+!\"\"*&F-F+%$_C1GF+F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 256 " " 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "dy/dx+tan(x)*y = x^3;" "6#/,& *&%#dyG\"\"\"%#dxG!\"\"F'*&-%$tanG6#%\"xGF'%\"yGF'F'*$F.\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "de := diff(y(x),x)+tan(x)*y(x)=x^3;\ndesolveLN(de,inf o=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6 #%\"xGF-\"\"\"*&-%$tanGF,F.F*F.F.*$)F-\"\"$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~ ~G/,&-%%DiffG6$-%\"yG6#%\"xGF,\"\"\"*&-%$tanGF+F-F)F-F-*$)F,\"\"$F-" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$-%$tanG6#%\"xGF-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G*&\"\"\"F%-%$co sG6#%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$cosG6#%\"xG!\"\"-%\"yGF'\"\"\"-%$IntG6$*&F( \"\"$F%F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&-%$cosG6#%\"xG!\"\"-%\"yGF'\"\"\",&-%$intG6$*&%#_u G\"\"$-F&6#F2F)/F2;%!GF(F,&%\"CG6#F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&-%$cosG F&\"\"\"-%$intG6$*&%#_uG\"\"$-F+6#F1!\"\"/F1;%!GF'F,F,*&&%\"CG6#F,F,F* F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"yG6#%\"xG,&*&-%$cosGF&\"\"\"-%$IntG6$*&F'\"\"$F*!\"\"F'F,F,*&F*F,% $_C1GF,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "subs(%,de);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%diffG6$,&*&-%$cosG6#%\"xG\"\"\"-%$IntG6$*&F-\"\"$ F*!\"\"F-F.F.*&F*F.%$_C1GF.F.F-F.*&-%$tanGF,F.F(F.F.*$)F-F3F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"xG\"\"$\"\"\"F$" }}}{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 "dx/dt+2*x = 5;" "6#/,&*&%#dxG\"\"\"%# dtG!\"\"F'*&\"\"#F'%\"xGF'F'\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "de := diff(x (t),t)+2*x(t)=5;\ndesolveLN(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"xG6#%\"tGF-\"\"\"*&\"\"#F.F*F.F. \"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$-%\"xG6#%\"tGF,\"\"\"*& \"\"#F-F)F-F-\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$\" \"#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G-%$ expG6#,$*&\"\"#\"\"\"%\"tGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$*&\"\"#\"\"\"%\"tGF+ F+F+-%\"xG6#F,F+-%$IntG6$,$*&\"\"&F+F%F+F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$*& \"\"#\"\"\"%\"tGF+F+F+-%\"xG6#F,F+,&*&#\"\"&F*F+F%F+F+&%\"CG6#F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"xG6#%\"tG,&#\"\"&\"\"#\"\"\"*&&%\"CG6#F,F,-%$expG6#,$*&F+F,F'F,! \"\"F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"x G6#%\"tG,&#\"\"&\"\"#\"\"\"*&-%$expG6#,$F'!\"#F,%$_C1GF,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 7" }}{PARA 256 "" 0 " " {TEXT -1 4 " " }{XPPEDIT 18 0 "dx/dt+2*x = exp(-2*t)*sin(t);" "6# /,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"\"#F'%\"xGF'F'*&-%$expG6#,$*&F+F'%\"tG F'F)F'-%$sinG6#F3F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "de := diff(x(t),t)+2*x(t)=e xp(-2*t)*sin(t);\ndesolveLN(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"xG6#%\"tGF-\"\"\"*&\"\"#F.F*F.F. *&-%$expG6#,$*&F0F.F-F.!\"\"F.-%$sinGF,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~ ~G/,&-%%DiffG6$-%\"xG6#%\"tGF,\"\"\"*&\"\"#F-F)F-F-*&-%$expG6#,$*&F/F- F,F-!\"\"F--%$sinGF+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~factor~.~.~~G-%$expG6#-% $IntG6$\"\"#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~ ~=~~~G-%$expG6#,$*&\"\"#\"\"\"%\"tGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$*&\"\"#\"\" \"%\"tGF+F+F+-%\"xG6#F,F+-%$IntG6$-%$sinGF/F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$*& \"\"#\"\"\"%\"tGF+F+F+-%\"xG6#F,F+,&-%$cosGF/!\"\"&%\"CG6#F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"xG6#%\"tG,&*&-%$expG6#,$*&\"\"#\"\"\"F'F0!\"\"F0-%$cosGF&F0F1*&& %\"CG6#F0F0F*F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "dsolve(de,x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&-%$expG6#,$F'!\"#\"\"\"-%$cosGF&F/!\" \"*&F*F/%$_C1GF/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "E xample 8" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "dx/dt+ 2*x = sin(t);" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"\"#F'%\"xGF'F'-%$sin G6#%\"tG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "de := diff(x(t),t)+2*x(t)=sin(t);\n desolveLN(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&- %%diffG6$-%\"xG6#%\"tGF-\"\"\"*&\"\"#F.F*F.F.-%$sinGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE ~.~.~~G/,&-%%DiffG6$-%\"xG6#%\"tGF,\"\"\"*&\"\"#F-F)F-F--%$sinGF+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$\"\"#%\"tG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G-%$expG6#,$%\"tG\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*&-%\"xG6#%\"tG\"\"\"-%$expG6#,$F(\"\"#F)-%$IntG6$*&-%$sinGF'F)F*F)F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"xG6#%\"tG\"\"\"-%$expG6#,$F(\"\"#F),(*&-%$cosGF'F)F*F)#! \"\"\"\"&*(#F.F5F)-%$sinGF'F)F*F)F)&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,( -%$cosGF&#!\"\"\"\"&*&#\"\"#F-\"\"\"-%$sinGF&F1F1*&&%\"CG6#F1F1-%$expG 6#,$F'!\"#F1F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "dsolve(de,x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,(-%$cosGF&#!\"\"\"\"&*&#\"\"#F-\"\"\"-%$sinGF&F1F 1*&-%$expG6#,$F'!\"#F1%$_C1GF1F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(%,de);\nsimplify(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%diffG6$,&*&-%$cosG6#%\"xG\"\"\" -%$IntG6$*&*$)F-\"\"$F.F.F*!\"\"F-F.F.*&F*F.%$_C1GF.F.F-F.*&-%$tanGF,F .F(F.F.*$F4F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"xG\"\"$\"\"\"F $" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 9" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dx/dt+2*x=sin(t)+ sin(3*t)/3+sin(5*t)/5+sin(7*t)/7" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\" \"#F'%\"xGF'F',*-%$sinG6#%\"tGF'*&-F/6#*&\"\"$F'F1F'F'F6F)F'*&-F/6#*& \"\"&F'F1F'F'F;F)F'*&-F/6#*&\"\"(F'F1F'F'F@F)F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "de := diff(x(t),t)+2*x(t)=sin(t)+sin(3*t)/3+sin(5*t)/5+sin(7*t)/7; \ndesolveLN(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/, &-%%diffG6$-%\"xG6#%\"tGF-\"\"\"*&\"\"#F.F*F.F.,*-%$sinGF,F.*&#F.\"\"$ F.-F36#,$F-F6F.F.*&#F.\"\"&F.-F36#,$F-F " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 10" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx+f(x)*y = g(x);" "6#/,&*&%#dyG\" \"\"%#dxG!\"\"F'*&-%\"fG6#%\"xGF'%\"yGF'F'-%\"gG6#F." }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "f := 'f': g := 'g':\nde := diff(y(x),x)+f(x)*y(x)=g(x);\ndesolveLN(de,info=true);\nfactor(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\" \"\"*&-%\"fGF,F.F*F.F.-%\"gGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$-% \"yG6#%\"xGF,\"\"\"*&-%\"fGF+F-F)F-F--%\"gGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~fa ctor~.~.~~G-%$expG6#-%$IntG6$-%\"fG6#%\"xGF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\"xG\"\"\"-%$expG6#-%$intG6$-%\"fG6#%#_uG/F 3;%!GF(F),&-F.6$*&-%\"gG6#%#_vGF)-F+6#-F.6$F0/F3;F6F>F)/F>F5F)&%\"CG6# F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&-%$expG6#,$-%$intG6$-%\"fG6#%#_uG/F4; %!GF'!\"\"\"\"\"-F/6$*&-%\"gG6#%#_vGF9-F+6#-F/6$F1/F4;F7F@F9/F@F6F9F9* &&%\"CG6#F9F9F*F9F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG* &-%$expG6#,$-%$intG6$-%\"fG6#%#_uG/F3;%!GF'!\"\"\"\"\",&-F.6$*&-%\"gG6 #%#_vGF8-F*6#-F.6$F0/F3;F6F@F8/F@F5F8&%\"CG6#F8F8F8" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"yG6#%\"xG*&,&-%$IntG6$*&-%\"gGF&\"\"\"-%$expG6#-F+6$-%\"fGF&F'F0F' F0%$_C1GF0F0-F26#-F+6$,$F6!\"\"F'F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolve LN" }{TEXT -1 30 ": particular solution examples" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examp le 1" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx+3*y = exp(2*x),y(0) = 2;" "6$/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"$F'%\"yGF'F' -%$expG6#*&\"\"#F'%\"xGF'/-F,6#\"\"!F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "de := d iff(y(x),x)+3*y(x)=exp(2*x);\nic := y(0)=2;\ndesolveLN(\{de,ic\},info= true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#% \"xGF-\"\"\"*&\"\"$F.F*F.F.-%$expG6#,$*&\"\"#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#%\"xGF,\"\"\"*&\"\"$F-F)F-F--%$expG6#,$*&\"\"#F -F,F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$\"\"$%\"xG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G-%$expG6#,$ *&\"\"$\"\"\"%\"xGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$*&\"\"$\"\"\"%\"xGF+F+F+ -%\"yG6#F,F+-%$IntG6$-F&6#,$*&\"\"&F+F,F+F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$*& \"\"$\"\"\"%\"xGF+F+F+-%\"yG6#F,F+,&*&#F+\"\"&F+-F&6#,$*&F3F+F,F+F+F+F +&%\"CG6#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~~G/&%\"CG6#\"\" \"#\"\"*\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"&F+-%$expG6#,$*&\"\"#F+ F'F+F+F+F+*&#\"\"*F,F+-F.6#,$*&\"\"$F+F'F+!\"\"F+F+" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{d e,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&*&# \"\"\"\"\"&F,-%$expG6#,$*&F-F,F'F,F,F,F,#\"\"*F-F,F,-F/6#,$*&\"\"$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 "x;" "6#%\"xG" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-y = x^4,y(1) = 3;" "6$/,&*&%#dyG \"\"\"%#dxG!\"\"F'%\"yGF)*$%\"xG\"\"%/-F*6#F'\"\"$" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "de := x*diff(y(x),x)-y(x)=x^4;\nic := y(1)=3;\ndesolveLN(\{de,ic\} ,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&*&%\"xG\"\"\" -%%diffG6$-%\"yG6#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#%\"xGF,\"\"\"*&F,!\"\"F)F-F/*$)F,\"\"$F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$,$*&\"\"\"F,%\"xG!\"\"F. F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G*&\"\"\"F %%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&%\"xG!\"\"-%\"yG6#F%\"\"\"-%$IntG6$*$)F%\"\"#F*F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"xG!\"\"-%\"yG6#F%\"\"\",&*&\"\"$F&F%F-F*&%\"CG6#F*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%EApplying~the~initial~condition~.~.~~G/&%\"CG6#\"\"\"#\"\")\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG,&*&#\"\"\"\"\"$F+*$)F'\"\"%F+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(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" yG6#%\"xG*&,&*$)F'\"\"$\"\"\"#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 3" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "dy/dx+2*x*y = exp(-3*x)*(2*x-3),y(0) = 2;" "6$/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"#F'%\"xGF'%\"yGF'F'*&-%$ex pG6#,$*&\"\"$F'F,F'F)F',&*&F+F'F,F'F'F4F)F'/-F-6#\"\"!F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "de := diff(y(x),x)+2*x*y(x)=exp(-3*x)*(2*x-3);\nic : = y(0)=2;\ndesolveLN(\{de,ic\},info=true):\nf := unapply(rhs(%),x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\" \"\"*(\"\"#F.F-F.F*F.F.*&-%$expG6#,$*&\"\"$F.F-F.!\"\"F.,&*&F0F.F-F.F. F7F8F." }}{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#%\"xGF,\"\"\"*(\"\"#F-F,F-F)F -F-,&*(F/F--%$expG6#,$*&\"\"$F-F,F-!\"\"F-F,F-F-*&F7F-F2F-F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Int egrating~factor~.~.~~G-%$expG6#-%$IntG6$,$*&\"\"#\"\"\"%\"xGF-F-F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G-%$expG6#*$)%\" xG\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#*$)%\"xG\"\"#\"\"\"F,-%\"yG6#F*F,-%$IntG 6$*&-F&6#,&*&\"\"$F,F*F,!\"\"F(F,F,,&*&F+F,F*F,F,F8F9F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$ex pG6#*$)%\"xG\"\"#\"\"\"F,-%\"yG6#F*F,,&-F&6#,&*&\"\"$F,F*F,!\"\"F(F,F, &%\"CG6#F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~~G/&%\"CG6#\"\" \"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%$expG6#,$* &\"\"$\"\"\"9$F3!\"\"F3-F.6#,$*$)F4\"\"#F3F5F3F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "plot(f(x),x=0..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)$\"\"#F)7 $$\"39LLLL3VfV!#>$\"3U*\\F\"e&4b(=!#<7$$\"3'pmm;H[D:)F/$\"3W=B2e:un#4$>\"F27$$\"3#pmmm@Xt=%F;$\"3P/([jx0 R7\"F27$$\"3QLLL3y_qXF;$\"3eeyM-9Hl5F27$$\"3i******\\1!>+&F;$\"3@B-&\\ rb;+\"F27$$\"3()******\\Z/NaF;$\"3y&)f&z')\\1S*F;7$$\"3'*******\\$fC&e F;$\"3K3^(*eujF))F;7$$\"3ELL$ez6:B'F;$\"3P@X.='\\SK)F;7$$\"3Smmm;=C#o' F;$\"3')y==_3`XxF;7$$\"3-mmmm#pS1(F;$\"3#fobXlhDF(F;7$$\"3]****\\i`A3v F;$\"3SrA:QL>UnF;7$$\"3slmmm(y8!zF;$\"3;!>C$>uo!H'F;7$$\"3V++]i.tK$)F; $\"31c_d,#=]\"eF;7$$\"39++](3zMu)F;$\"3EwW\\$*eb\"Q&F;7$$\"3#pmm;H_?<* F;$\"3KAx.cA*)\\\\F;7$$\"3emm;zihl&*F;$\"3YI+;yNHsXF;7$$\"39LLL3#G,*** F;$\"3?Azp%e3a=%F;7$$\"371QF;7$$\"3!****\\PQ#\\\" 3\"F2$\"3[awuS0s%\\$F;7$$\"3BLL$e\"*[H7\"F2$\"3[QKH9z&z<$F;7$$\"3#**** ***pvxl6F2$\"3_6t)R3g=(GF;7$$\"3z****\\_qn27F2$\"3m(pdlg-Hf#F;7$$\"3%) ***\\i&p@[7F2$\"3uE&\\\\^5>M#F;7$$\"3#)****\\2'HKH\"F2$\"3+nlPe-X%3#F; 7$$\"3_mmmwanL8F2$\"3)p?F.5g:(=F;7$$\"3'******\\2goP\"F2$\"3i(p')yR,Gm \"F;7$$\"3CLLeR<*fT\"F2$\"3Gq*)y+0\\*[\"F;7$$\"3'******\\)Hxe9F2$\"3zW <(*)QXkJ\"F;7$$\"3Ymm\"H!o-*\\\"F2$\"3]&H'3Wx[o6F;7$$\"3))***\\7k.6a\" F2$\"3Dm!z2xY$G5F;7$$\"3emmmT9C#e\"F2$\"3#)okzOo#z/*F/7$$\"3\"****\\i! 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together with the analytical solution.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "ff := dsolve(\{de,ic\},y(x),type=numeric,output =listprocedure);\ngn := subs(ff,y(x)):\nplot3 := plot('gn'(x),x=1..3,y =0..3,color=black):\ndisplay([plot3,plot2],labels=[`x`,`y(x)`]);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$ $\"\"\"\"\"!$\"\"#F*7$$\"+3VfV5!\"*$\"+cWRZ=F07$$\"+$[D:3\"F0$\"+\\sLH 6F07$$\"+AXt=9F0$\"+]-,g5 F07$$\"+\"y_qX\"F0$\"+g\\f65F07$$\"+l+>+:F0$\"+%=&*Rh*!#57$$\"+vW]V:F0 $\"+s'z4:*Fdo7$$\"+NfC&e\"F0$\"+OO[R()Fdo7$$\"+!=^Ji\"F0$\"+%\\\\BR)Fd o7$$\"+#=C#o;F0$\"+fP>4!)Fdo7$$\"+FpS1F0$\"+vGyUjFdo7$$\"+G;cc>F0$\"+\"QR O8'Fdo7$$\"+@G,**>F0$\"+r1/@fFdo7$$\"+!o2J/#F0$\"+%f&Fdo 7$$\"+`qn2AF0$\"+-)\\%Q]Fdo7$$\"+cp@[AF0$\"+$>tH*[Fdo7$$\"+3'HKH#F0$\" +EMoRZFdo7$$\"+xanLBF0$\"+-<\")3YFdo7$$\"+v+'oP#F0$\"+d*RdZ%Fdo7$$\"+S <*fT#F0$\"+94qgVFdo7$$\"+&)HxeCF0$\"+y,cSUFdo7$$\"+.o-*\\#F0$\"+z(RD8% Fdo7$$\"+TO5TDF0$\"+co\\CSFdo7$$\"+U9C#e#F0$\"+ufPBRFdo7$$\"+1*3`i#F0$ 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-1 70 "Another way to solve this initial value p roblem is to make use of the " }{TEXT 263 23 "Heaviside step function " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "H (x)=PIECEWISE([0,x<0],[undefined,x=0],[1,x>0])" "6#/-%\"HG6#%\"xG-%*PI ECEWISEG6%7$\"\"!2F'F,7$%*undefinedG/F'F,7$\"\"\"2F,F'" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "Maple has the Heaviside function buil t-in as " }{TEXT 268 12 "Heaviside(x)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "'H'(x)= convert(Heaviside(x),piecewise);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"HG6#%\"xG-%*PIECEWISEG6%7$\"\"!2F'F,7$%*undefinedG/F'F,7$\"\"\"2F,F '" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "We \+ can redefine " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 50 " \+ in terms of the Heaviside function (except where " }{XPPEDIT 18 0 "x=1 " "6#/%\"xG\"\"\"" }{TEXT -1 14 ") as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "f := x -> p iecewise(x<1,1,0):\n'f(x)'=f(x);\nalias(H=Heaviside):\nf := unapply(co nvert(f(x),Heaviside),x):\n'f(x)'=f(x);\nplot(f(x),x=0..3,thickness=2) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\" \"\"2F'F,7$\"\"!%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"f G6#%\"xG,&\"\"\"F)-%\"HG6#,&F'F)F)!\"\"F." }}{PARA 13 "" 1 "" {GLPLOT2D 511 168 168 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"\"!F)$\"\"\"F )7$$\"3s******\\i9Rl!#>F*7$$\"3/++vVA)GA\"!#=F*7$$\"3+++]Peui=F3F*7$$ \"3A++]i3&o]#F3F*7$$\"3%)***\\(oX*y9$F3F*7$$\"3z***\\P9CAu$F3F*7$$\"3! )***\\P*zhdVF3F*7$$\"31++v$>fS*\\F3F*7$$\"3$)***\\(=$f%GcF3F*7$$\"3Q++ +Dy,\"G'F3F*7$$\"33++]7&)\\\" FcpF(7$$\"3)***\\P>:mk:FcpF(7$$\"3'***\\iv&QAi\"FcpF(7$$\"31++vtLU%o\" FcpF(7$$\"3!******\\Nm'[FcpF(7$$\"3z*****\\@80+#FcpF(7$$\"31++]7,Hl ?FcpF(7$$\"3()**\\P4w)R7#FcpF(7$$\"3;++]x%f\")=#FcpF(7$$\"3!)**\\P/-a[ AFcpF(7$$\"3/+](=Yb;J#FcpF(7$$\"3')****\\i@OtBFcpF(7$$\"3')**\\PfL'zV# FcpF(7$$\"3>+++!*>=+DFcpF(7$$\"3-++DE&4Qc#FcpF(7$$\"3=+]P%>5pi#FcpF(7$ $\"39+++bJ*[o#FcpF(7$$\"33++Dr\"[8v#FcpF(7$$\"3++++Ijy5GFcpF(7$$\"31+] P/)fT(GFcpF(7$$\"31+]i0j\"[$HFcpF(7$$\"\"$F)F(-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fiw-%%VIEWG6$;F(F hv%(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 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Converting " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 47 " back to piecewise form indicates that the new " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 43 " is not defined at the discontinu ity where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 61 "However, this does not affect the calcula tion significantly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "convert(f(x),piecewise,x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6%7$\"\"\"2%\"xGF'7$%*undefinedG/F) F'7$\"\"!2F'F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Here is the alternative solution. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "de := diff(y(x),x)+y(x) = f(x);\nic := y(0)=0 ;\ndesolveLN(\{de,ic\},info=true);\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"F*F. ,&F.F.-%\"HG6#,&F-F.F.!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ic G/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$-%\"yG6#%\"xGF,\" \"\"F)F-,&F-F--%\"HG6#,&F,F-F-!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~factor~.~.~~G- %$expG6#-%$IntG6$\"\"\"%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~ ~~~~~~~~~~~~=~~~G-%$expG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\"xG\"\"\"-%$expGF'F)- %$IntG6$,&F*F)*&F*F)-%\"HG6#,&F(F)F)!\"\"F)F5F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\"xG \"\"\"-%$expGF'F),*F*F)*&F*F)-%\"HG6#,&F(F)F)!\"\"F)F2*&F.F)-F+6#F)F)F )&%\"CGF5F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~~G/&%\"CG6#\"\" \"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,*\"\"\"F)-%\"HG6#,&F'F)F)!\"\"F.*&-%$exp G6#,&F'F.F)F)F)F*F)F)-F16#,$F'F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,*\"\"\"F--%\"HG6#,&9$F-F-!\" \"F3*&-%$expG6#,&F2F3F-F-F-F.F-F--F66#,$F2F3F3F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 158 "Converting the new so lution to piecewise form shows that this is essentially the same solut ion as before (except that the solution is not defined when x = 1)." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "convert(g(x),piecewise);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*P IECEWISEG6%7$,&-%$expG6#,$%\"xG!\"\"F-\"\"\"F.2F,F.7$,&-F)6#F-F-%*unde finedGF./F,F.7$,&F(F--F)6#,&F,F-F.F.F.2F.F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The graph of the new solution i s the same as before." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(g(x),x=0..3,y=0..1,color=coral,thick ness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'C URVESG6#7W7$$\"\"!F)F(7$$\"3')*****\\7t&pK!#>$\"3kt<7(Q+n@$F-7$$\"3s** ****\\i9RlF-$\"37T%Q/;H*HjF-7$$\"33++vVV)RQ*F-$\"3=fS*\\F=$\"35\"*y?^!*3JRF=7$$\"3$)***\\(=$f%GcF=$\"3X(y7AZUTI% F=7$$\"3Q+++Dy,\"G'F=$\"3gfO$yjiRm%F=7$$\"33++]7&)\\\"Fcq$\"32wz'eZ'oRQF=7$$\"3)*** \\P>:mk:Fcq$\"3K*[O#3w$Rf$F=7$$\"3'***\\iv&QAi\"Fcq$\"3)Hrhzr`GR$F=7$$ \"31++vtLU%o\"Fcq$\"3?g\"\\8!fH)=$F=7$$\"3!******\\Nm'[Fcq$\"3#=D%[rIipCF=7$$\"3z*****\\ @80+#Fcq$\"3PtV/M%[UK#F=7$$\"31++]7,Hl?Fcq$\"3WC5!3bj%y@F=7$$\"3()**\\ P4w)R7#Fcq$\"33!yBfmtU0#F=7$$\"3;++]x%f\")=#Fcq$\"3k_#[:#yeE>F=7$$\"3! )**\\P/-a[AFcq$\"3Zbk#[U,P\"=F=7$$\"3/+](=Yb;J#Fcq$\"3\"p-,\"[ow-aI')3(3g\"F=7$$\"3')**\\PfL'zV#Fcq$\"3rR43%R A2]\"F=7$$\"3>+++!*>=+DFcq$\"3_n<^]\\>59F=7$$\"3-++DE&4Qc#Fcq$\"3&=P+& *oiKK\"F=7$$\"3=+]P%>5pi#Fcq$\"3N()pc&zVBC\"F=7$$\"39+++bJ*[o#Fcq$\"3m !QcMjdB<\"F=7$$\"33++Dr\"[8v#Fcq$\"3/B7&p!4)p4\"F=7$$\"3++++Ijy5GFcq$ \"3NsH=7$yO.\"F=7$$\"31+]P/)fT(GFcq$\"3j45`94.-(*F-7$$\"31+]i0j\"[$HFc q$\"3m;X!p6K58*F-7$$\"\"$F)$\"35v[(o[@[b)F--%*THICKNESSG6#\"\"#-%'COLO URG6&%$RGBG$\"*++++\"!\")$\")AR!)\\Fe\\lF(-%+AXESLABELSG6$Q\"x6\"Q\"yF \\]l-%%VIEWG6$;F(Fg[l;F($\"\"\"F)" 1 2 0 1 10 2 2 6 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 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 8" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "dx/dt+2*x=sin(t)+sin(3*t)/3+sin(5*t)/5+sin(7*t)/7" "6#/ ,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"\"#F'%\"xGF'F',*-%$sinG6#%\"tGF'*&-F/6# *&\"\"$F'F1F'F'F6F)F'*&-F/6#*&\"\"&F'F1F'F'F;F)F'*&-F/6#*&\"\"(F'F1F'F 'F@F)F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "f := t -> sin(t)+sin(3*t)/3+sin(5* t)/5+sin(7*t)/7;\nde := diff(x(t),t)+2*x(t)=f(t);\nic := x(0)=0;\ndeso lveLN(\{de,ic\},info=true):\ng := unapply(rhs(%),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,*-%$sinG6# 9$\"\"\"*&#F1\"\"$F1-F.6#,$F0F4F1F1*&#F1\"\"&F1-F.6#,$F0F:F1F1*&#F1\" \"(F1-F.6#,$F0F@F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/, &-%%diffG6$-%\"xG6#%\"tGF-\"\"\"*&\"\"#F.F*F.F.,*-%$sinGF,F.*&#F.\"\"$ F.-F36#,$F-F6F.F.*&#F.\"\"&F.-F36#,$F-F%#icG/-%\"xG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Lin ear~DE~.~.~~G/,&-%%DiffG6$-%\"xG6#%\"tGF,\"\"\"*&\"\"#F-F)F-F-,*-%$sin GF+F-*&#F-\"\"$F--F26#,$F,F5F-F-*&#F-\"\"&F--F26#,$F,F;F-F-*&#F-\"\"(F --F26#,$F,FAF-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$\" \"#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G-%$ expG6#,$%\"tG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"xG6#%\"tG\"\"\"-%$expG6#,$F(\"\"#F)-%$In tG6$,$*&,*-%$sinGF'\"$0\"*&\"#NF)-F66#,$F(\"\"$F)F)*&\"#@F)-F66#,$F(\" \"&F)F)*&\"#:F)-F66#,$F(\"\"(F)F)F)F*F)#F)F7F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&-%\"xG6#%\"tG \"\"\"-%$expG6#,$F(\"\"#F),4*&-%$cosGF'F)F*F)#!\"\"\"\"&*(#F.F5F)F*F)- %$sinGF'F)F)*&#F)\"#8F)*&-F26#,$F(\"\"$F)F*F)F)F4*(#F.\"#RF)F*F)-F9F?F )F)*&#F)\"#HF)*&F*F)-F26#,$F(F5F)F)F4*(#F.\"$X\"F)F*F)-F9FKF)F)*&#F)\" #`F)*&F*F)-F26#,$F(\"\"(F)F)F4*(#F.\"$r$F)F*F)-F9FVF)F)&%\"CG6#F)F)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%EApplying~the~initial~condition~.~.~~G/&%\"CG6#\"\"\"#\"&'*H$\"&0** *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,4-%$cosG6#9$#!\"\"\"\" &*&#\"\"#F3\"\"\"-%$sinGF/F7F7*&#F7\"#8F7-F.6#,$F0\"\"$F7F2*&#F6\"#RF7 -F9F>F7F7*&#F7\"#HF7-F.6#,$F0F3F7F2*&#F6\"$X\"F7-F9FIF7F7*&#F7\"#`F7-F .6#,$F0\"\"(F7F2*&#F6\"$r$F7-F9FSF7F7*&#\"&'*H$\"&0***F7-%$expG6#,$F0! 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3 " " }{XPPEDIT 18 0 "dy/dx+f(x)*y = g(x),y(x0) = y0;" "6 $/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"fG6#%\"xGF'%\"yGF'F'-%\"gG6#F./-F/6 #%#x0G%#y0G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "f := 'f': g := 'g':\nde := diff(y( x),x)+f(x)*y(x)=g(x);\nic := y(x0)=y0;\nsol := desolveLN(\{de,ic\},inf o=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6 #%\"xGF-\"\"\"*&-%\"fGF,F.F*F.F.-%\"gGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#%#x0G%#y0G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$-% \"yG6#%\"xGF,\"\"\"*&-%\"fGF+F-F)F-F--%\"gGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~fa ctor~.~.~~G-%$expG6#-%$IntG6$-%\"fG6#%\"xGF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\"xG\"\"\"-%$expG6#-%$intG6$-%\"fG6#%#_uG/F 3;%!GF(F),&-F.6$*&-%\"gG6#%#_vGF)-F+6#-F.6$F0/F3;F6F>F)/F>F5F)&%\"CG6# F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~~G/&%\"CG6#\"\"\", &*&%#y0GF(-%$expG6#-%$IntG6$-%\"fG6#%\"xG/F5;%!G%#x0GF(F(-F06$*&-%\"gG 6#%#_vGF(-F-6#-%$intG6$-F36#%#_uG/FH;F8F@F(/F@F7!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/-% \"yG6#%\"xG*&,&-%$IntG6$*&-%\"gG6#%#_vG\"\"\"-%$expG6#-%$intG6$-%\"fG6 #%#_uG/F>;%!GF3F4/F3;%#x0GF)F4*&%#y0GF4-F66#-F-6$-F;FAF)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simpl ify(subs(sol,de));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xGF$ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Howe ver, Maple's solution does not check in the differential equation." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "sol2 := dsolve(\{de,ic\},y(x));\nsimplify(subs(sol2,de));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G/-%\"yG6#%\"xG*&,&-%$IntG6$*&- %\"gG6#%\"uG\"\"\"-%$expG6#-F-6$-%\"fGF2F3F4/F3;%#x0GF)F4%#y0GF4F4-F66 #-F-6$,$F:!\"\"F " 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 "T asks" }}{PARA 0 "" 0 "" {TEXT -1 59 "In questions 1 to 8, solve the gi ven initial value problem." }}{PARA 0 "" 0 "" {TEXT -1 116 "In each ca se plot the graph of the solution along with the associated direction \+ field for the differential equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 4 "Note" }{TEXT -1 67 ": When setting up t he differential equation to be solved by either " }{TEXT 0 6 "dsolve" }{TEXT -1 4 " or " }{TEXT 0 9 "desolveLN" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 4 "Use " }{TEXT 0 12 "diff(y(x),x)" }{TEXT -1 37 " to replace the mathematical symbol " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG \"\"\"%#dxG!\"\"" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 18 "Repl ace y by y(x) " }{TEXT 265 49 "everywhere y appears in the differentia l equation" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "dy/dx+y \+ = sin(x),y(0) = 0;" "6$/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF'-%$sinG6#%\" xG/-F*6#\"\"!F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "______ _______________________________" }}{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 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 5 " " } {XPPEDIT 18 0 "dy/dx+x*y/(x^2+9) = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F' *(%\"xGF'%\"yGF',&*$F+\"\"#F'\"\"*F'F)F'\"\"!" }{TEXT -1 4 ", " } {XPPEDIT 18 0 "y(0) = 1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "dy/dx+2*x*y = x;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"#F'%\"xGF'%\"yGF'F'F," }{TEXT -1 4 ", " }{XPPEDIT 18 0 "y(0) = 2" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "_________________________________ ____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 270 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y = x*sin(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\" F'%\"yGF'*&%\"xGF'-%$sinG6#F,F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(P i/2) = 1" "6#/-%\"yG6#*&%#PiG\"\"\"\"\"#!\"\"F)" }{TEXT -1 3 " " }} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {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 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }} {PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "``(1+x);" "6#-%!G6#, &\"\"\"F'%\"xGF'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-x*y = x^2;" "6 #/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"xGF'%\"yGF'F)*$F+\"\"#" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "_________________________________ ____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "``(1+x^2)" "6 #-%!G6#,&\"\"\"F'*$%\"xG\"\"#F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx +x*y+x+x^3 = 0;" "6#/,**&%#dyG\"\"\"%#dxG!\"\"F'*&%\"xGF'%\"yGF'F'F+F' *$F+\"\"$F'\"\"!" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "y(0) = 2" "6#/-% \"yG6#\"\"!\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "_____ ________________________________" }}{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 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }}{PARA 0 "" 0 "" {TEXT -1 4 " " } {TEXT 272 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+(1+x)*y = exp(-x )*sin(2*x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&,&F'F'%\"xGF'F'%\"yGF'F' *&-%$expG6#,$F,F)F'-%$sinG6#*&\"\"#F'F,F'F'" }{TEXT -1 14 ", y(1) = \+ 0 " }}{PARA 0 "" 0 "" {TEXT -1 37 "__________________________________ ___" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q8" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "``(1+exp(x)); " "6#-%!G6#,&\"\"\"F'-%$expG6#%\"xGF'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+exp(x)*y = x^4;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%$expG6#%\"x GF'%\"yGF'F'*$F.\"\"%" }{TEXT -1 13 ", y(0) = 2 " }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{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 37 "____________ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q9" }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "f(x) = PIECEWISE([x, x <= 2],[0, \+ 2 < x]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$F'1F'\"\"#7$\"\"!2F-F'" } {TEXT -1 72 ", find a continuous solution which satisfies the differe ntial equation " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "dy/dx+2*x*y = f(x) ;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"#F'%\"xGF'%\"yGF'F'-%\"fG6#F, " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 28 " for all positive va lues of " }{TEXT 271 1 "x" }{TEXT -1 24 ", except possibly where " } {XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 42 ", and is subject \+ to the initial condition " }{XPPEDIT 18 0 "y(0) = -1;" "6#/-%\"yG6#\" \"!,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "____ _________________________________" }}{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 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q10 " }}{PARA 0 "" 0 "" {TEXT -1 68 "(a) C onstruct a first order linear differential equation, for which " } {TEXT 0 6 "dsolve" }{TEXT -1 4 " or " }{TEXT 0 9 "desolveLN" }{TEXT -1 31 " can find the general solution." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 136 "(b) For the differential equation ch osen in (a) pick an initial condition and find the particular solution with this initial condition. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 119 "(c) Plot the graph of the particular sol ution along with the direction field associated with the differential \+ equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{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 37 "____________________ _________________" }}{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 }