{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 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 264 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasi s" -1 265 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 257 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 257 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 269 "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 "Hea ding 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 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 "T imes" 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 } } {SECT 0 {PARA 3 "" 0 "" {TEXT -1 45 "A procedure for solving 1st order exact DE's " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, \+ B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 20 "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 269 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 46 "A procedure for solving 1st order exact DE's: " }{TEXT 0 9 "desolveEX" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "desolv eEX: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 261 2 " " }{TEXT -1 18 " desolveEX( de )" }}{PARA 0 "" 0 "" {TEXT -1 25 " desolveEX( \{de,ic\} )" }}{PARA 0 "" 0 "" {TEXT -1 31 " \+ desolveEX( \{de,ic\}, y(x) )" }{TEXT 262 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 t he form diff(y(x),x)," }}{PARA 0 "" 0 "" {TEXT -1 94 " \+ (if x and y are the independent and dependent variables respec tively)." }}{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 "desolveEX" }{TEXT -1 22 " att empts to solve an " }{TEXT 264 5 "exact" }{TEXT -1 41 " first order li near differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M(x,y)+N(x,y);" "6#,&-%\"MG 6$%\"xG%\"yG\"\"\"-%\"NG6$F'F(F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/d x = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Diff(M(x,y),y) = Diff(N(x,y ),x);" "6#/-%%DiffG6$-%\"MG6$%\"xG%\"yGF+-F%6$-%\"NG6$F*F+F*" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 44 "by constructing a function of \+ two variables " }{XPPEDIT 18 0 "F(x,y);" "6#-%\"FG6$%\"xG%\"yG" } {TEXT -1 11 " such that " }{XPPEDIT 18 0 "Diff(F(x,y),x) = M(x,y);" "6 #/-%%DiffG6$-%\"FG6$%\"xG%\"yGF*-%\"MG6$F*F+" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "Diff(F(x,y),y) = N(x,y);" "6#/-%%DiffG6$-%\"FG6$%\"xG% \"yGF+-%\"NG6$F*F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 22 "by \+ one of two methods:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 266 8 "Method 1" }{TEXT -1 18 " (standard scheme)" }}{PARA 0 " " 0 "" {TEXT -1 6 " Let " }{XPPEDIT 18 0 "F(x,y) = F[1](x,y)+g(y);" " 6#/-%\"FG6$%\"xG%\"yG,&-&F%6#\"\"\"6$F'F(F--%\"gG6#F(F-" }{TEXT -1 8 " , where " }{XPPEDIT 18 0 "F[1](x,y) = Int(M(x,y),x);" "6#/-&%\"FG6#\" \"\"6$%\"xG%\"yG-%$IntG6$-%\"MG6$F*F+F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "Choose " }{XPPEDIT 18 0 "g(y)" "6#-%\"gG6#%\"yG" } {TEXT -1 9 " so that " }{XPPEDIT 18 0 "Diff(F(x,y),y) = N(x,y);" "6#/- %%DiffG6$-%\"FG6$%\"xG%\"yGF+-%\"NG6$F*F+" }{TEXT -1 4 " by " } {XPPEDIT 18 0 "g(y) = Int(``(N(x,y)-diff(F[1](x,y),y)),y);" "6#/-%\"gG 6#%\"yG-%$IntG6$-%!G6#,&-%\"NG6$%\"xGF'\"\"\"-%%diffG6$-&%\"FG6#F36$F2 F'F'!\"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 8 "Method 2" }{TEXT -1 21 " (alternative scheme)" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "F(x,y) = F[2](x,y )+h(x);" "6#/-%\"FG6$%\"xG%\"yG,&-&F%6#\"\"#6$F'F(\"\"\"-%\"hG6#F'F/" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "F[2](x,y) = Int(N(x,y),y);" "6# /-&%\"FG6#\"\"#6$%\"xG%\"yG-%$IntG6$-%\"NG6$F*F+F+" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 7 "Choose " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6# %\"xG" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "Diff(F(x,y),y) = M(x,y) ;" "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF+-%\"MG6$F*F+" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "h(x) = Int(``(M(x,y)-diff(F[2](x,y),x)),x);" "6#/-%\"h G6#%\"xG-%$IntG6$-%!G6#,&-%\"MG6$F'%\"yG\"\"\"-%%diffG6$-&%\"FG6#\"\"# 6$F'F2F'!\"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 48 "In both cases the general solution has th e form " }{XPPEDIT 18 0 "F(x,y) = C[1];" "6#/-%\"FG6$%\"xG%\"yG&%\"CG6 #\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 63 "If an initial condition is provided, the value of \+ the constant " }{XPPEDIT 18 0 "C[1];" "6#&%\"CG6#\"\"\"" }{TEXT -1 15 " will be found." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 93 "info=true/false\nThe option info=true causes intermediate steps in th e solution to be printed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 24 "format=explicit/implicit" }}{PARA 0 "" 0 "" {TEXT -1 96 "When format=explicit, which is the default option, an attempt i s made to solve for y explicitly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 43 "scheme=std/alt or scheme=standard/alterna te" }}{PARA 0 "" 0 "" {TEXT -1 144 "When \"scheme=std\", or \"scheme= standard\", method 1 above is attempted, otherwise, if \"scheme=alt\", or \"scheme=alternate\", method 2 is attempted." }}{PARA 0 "" 0 "" {TEXT -1 75 "The default is \"scheme=std\", but if this fails, method \+ 2 will be attempted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 16 "How to activate:" } {TEXT 256 1 "\n" }{TEXT -1 155 "To make the procedure active open the \+ subsection, place the cursor anywhere after the prompt [ > and press \+ [Enter].\nYou can then close up the subsection. " }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 25 "desolveEX: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12418 "desolveEX := proc()\n \n local ff,vars,derivs,x,yx,y,df,drv,prntflg,Options ,lsde,sgny0,\n j,intg,val,soln,initcond,de,ic,x0,y0,rsic,lsic, const,\n Mxy,Nxy,g,h,dMxy,dNxy,F1,gpy,gy,hpx,hx,F,Fxy,Rxy,la,N ,M,\n goodsols,removeconsts,sols,frmt,assignedconst,i,schm,\n \+ startopts,xx,yy,ee,cc,sol,yy0,tst,Cs,gt,jS,j1,s,nvars;\n glo bal C;\n \n # Remove any solutions involving \"RootOf\".\n goods ols := proc(sol::list)\n local goodlist,i;\n goodlist := NUL L;\n for i from 1 to nops(sol) do\n if indets(sol[i],'spe cfunc(anything,RootOf)')=\{\} then\n goodlist := goodlist,s ol[i];\n end if;\n end do;\n [goodlist]; \+ \n end proc: # of goodsols\n\n # Remove any constants from a sum o f terms\n removeconsts := proc(expr::algebraic)\n local terms,s um,i;\n if op(0,expr)<>`+`then return expr end if;\n terms : = [op(expr)];\n sum := 0;\n for i from 1 to nops(terms) do\n if not type(terms[i],complexcons) then\n sum := su m + terms[i];\n end if;\n end do;\n sum;\n end pro c: # of removeconsts\n \n # start of main procedure\n if nargs>0 then \n ff := args[1]\n else\n error \"at least one argum ent must be supplied\"\n end if;\n initcond := false;\n if type( ff,\{set(equation),list(equation)\}) and nops(ff)=2 then\n ff := \+ map(_u -> if has(_u,D) then convert(_u,diff) else _u end if,ff);\n \+ de := op(1,ff);\n ic := op(2,ff);\n if not has(de,diff) th en\n de := op(2,ff);\n ic := op(1,ff);\n end if; \n initcond := true;\n elif type(ff,equation) then\n if ha s(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 eq uation or a set (or list) of 2 equations\",ff;\n end if;\n\n start opts := 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 argu ment, %1, has incorrect form for the dependent variable\",ee;\n \+ end if;\n end if;\n end if;\n\n prntflg := false;\n frmt \+ := 'explicit';\n schm='std';\n if nargs>1 then\n Options:=[ar gs[startopts..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 i f;\n if hasoption(Options,'format','frmt','Options') then\n \+ if not (frmt='explicit' or frmt='implicit') then\n error \"\\\"format\\\" must be 'explicit' or 'implicit'\"\n end if; \n end if;\n if hasoption(Options,'scheme','schm','Options') then\n if not member(schm,\{'std','standard','alt','alternate '\}) then\n error \"\\\"scheme\\\" must be 'standard','std' ,'alternate' or 'alt'\"\n end if;\n if schm='standard' then schm := 'std' \n elif schm='alternate' then schm := 'alt ' end if;\n end if;\n if nops(Options)>0 then\n erro r \"%1 is not a valid option for %2\",op(1,Options),procname;\n e nd if;\n end if;\n\n derivs := indets(de,'specfunc(anything,diff)' );\n if derivs=\{\} then\n error \"the 1st argument, %1, is inv alid .. it should be a differential equation or a set (or list) contai ning a differential equation 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 indepe ndent variable occurring in the derivative(s)\";\n else\n \+ error \"there should only be one independent variable in the differen tial equation\"\n end if;\n end if;\n nvars := nops(indets(de rivs,'anyfunc(name)'));\n if nvars<>1 then\n if nvars=0 then\n \+ error \"there is a problem with the dependent variable occurri ng in the derivative(s)\"\n else\n error \"there should o nly be one dependent variable in the differential equation\"\n en d if;\n end if;\n\n if nops(derivs)<>1 then\n error \"there a re too many derivatives in the differential equation .. note that the \+ differential equation must be of order 1\"\n end if;\n df := op(1, derivs);\n if type(df,function) and op(0,df)=diff and nops(df)=2 the n\n yx := op(1,df);\n if not type(yx,anyfunc(name)) then\n \+ error \"the 1st argument %1, in the derivative, %2, is invalid \+ .. it should be the 'unknown' dependent variable\",yx,df;\n end i f; \n x := op(2,df);\n if not type(x,name) then\n er ror \"the 2nd argument %1, in the derivative, %2, is invalid .. it sho uld be the independent variable\",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 vars := indets(de,name);\n if member(y,vars) then\n error \"%1 and %2 cannot both appear in the differential \+ equation\",yx,y;\n end if;\n if op(1,yx)<>x then\n error \"th e derivative, %1, does not make sense\",df;\n end if;\n\n if start opts=3 then \n if x<>xx or y<>yy then\n error \"cannot so lve the differential equation for %1\",ee;\n end if;\n end if; \n \n if initcond then\n lsic := lhs(ic);\n if type(lsic,f unction) 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 decipherable\"\n end if;\n \+ rsic := rhs(ic);\n if type(rsic,algebraic) then\n y0 \+ := rsic;\n if has(y0,\{x,y\}) then\n error \"initia l condition must not involve %1 or %2\",x,y;\n end if;\n \+ else\n error \"initial condition is not decipherable\"\n \+ end if;\n end if;\n\n if rhs(de)=0 then\n lsde := lhs(de);\n \+ else\n lsde := lhs(de)-rhs(de);\n end if;\n if patmatch(lsd e,conditional(M::algebraic+\n N::algebra ic*diff(yx,x),\n indets(M,'specfunc(anything,diff)')=\{\} and \n \+ indets(N,'specfunc(anything,diff)')=\{\}),'la') then\n Mxy : = subs(yx=y,subs(la,M));\n Nxy := subs(yx=y,subs(la,N));\n elif patmatch(lsde,conditional(diff(yx,x)+R::algebraic,\n indets(R,'s pecfunc(anything,diff)')=\{\}),'la') then\n Rxy := subs(yx=y,subs (la,R));\n Mxy := numer(Rxy);\n Nxy := denom(Rxy);\n else \n drv := solve(de,df);\n if nops([drv])=1 then\n Rx y := subs(yx=y,drv);\n Mxy := -numer(Rxy);\n Nxy := de nom(Rxy);\n else\n error \"cannot extract required terms \+ from the ODE\"\n end if;\n end if; \n\n dMxy := diff(Mxy,y) ;\n dNxy := diff(Nxy,x);\n\n tst := testeq(dMxy=dNxy);\n if not \+ tst or tst=FAIL then\n error \"the DE is not exact\"\n end if; \n \n if prntflg then\n print(``);\n print(`Test for exa ctness . .`);\n print(Diff(Mxy,y)=dMxy);\n print(Diff(Nxy,x) =dNxy);\n print(``);\n print(`Exact DE . . `,Mxy + (Nxy)*``( dy/dx)=0);\n print(``);\n end if;\n\n if assigned(C) and not \+ type(eval(C),table) then\n C := table();\n WARNING(\"C has b een redefined as a table for use as arbitrary constants\");\n end if ;\n # Find the indices j1,j2 to use in the constants\n Cs := selec t(type,indets(de),'specindex(posint,C)');\n gt := proc(_u) local s,j ;\n typematch(_u,C[j::posint],'s'); \n subs(s,j) \n end proc:\n jS := sort([op(map(gt,Cs))]);\n for i to no ps(jS)+1 do\n if not member(i,jS) then j1 := i; break; end if; \+ \n end do;\n\n if schm=alt then goto(2222) end if;\n\n intg := I nt(Mxy,x);\n if prntflg then\n print(`Let `,F(x,y)=simplify(int g)+g(y));\n fi;\n val := value(intg);\n if indets(val,'specfunc( anything,int)')=\{\} then \n F1 := simplify(combine(val,trig));\n else\n if prntflg then\n print(`cannot find `,intg);pr int(``);\n end if;\n goto(2222);\n end if;\n if prntflg \+ then\n print(` = `,F1+g(y));print(``);\n end if;\n \n \+ gpy := simplify(combine(Nxy-diff(F1,y),trig));\n if has(gpy,x) then \n if prntflg then\n print(gpy,`should not depend on`,x); print(``);\n end if;\n goto(2222);\n end if;\n if prntfl g then\n print(`where `,Diff(g(y),y)=Nxy-Diff(F1,y));\n prin t(` =`,gpy);print(``);\n end if;\n intg := Int(gpy,y);\n \+ val := value(intg);\n if indets(val,'specfunc(anything,int)')=\{\} then\n gy := simplify(combine(val,trig));\n else\n if prn tflg then\n print(`cannot find `,intg);print(``);\n end i f;\n goto(2222);\n end if;\n if prntflg then\n print(`so `,g(y)=intg);\n print(` = `,gy);print(``);\n end if;\n Fx y := removeconsts(F1+gy);\n if prntflg then\n print(F(x,y)=Fxy, ` satisfies . .`);\n print(Diff(F(x,y),x)=Mxy);\n print(Dif f(F(x,y),y)=Nxy);print(``);\n end if;\n soln := Fxy=C[j1];\n got o(1111);\n\n 2222:\n\n intg := Int(Nxy,y);\n if prntflg then\n \+ print(`Let `,F(x,y)=simplify(intg)+h(x));\n end if;\n val := v alue(intg);\n if indets(val,'specfunc(anything,int)')=\{\} then \n \+ F1 := simplify(combine(val,trig));\n else\n if prntflg then \n print(`cannot find `,intg);print(``);\n end if;\n \+ return NULL;\n end if;\n if prntflg then\n print(` = `,F1 +h(x));print(``);\n end if;\n \n hpx := simplify(combine(Mxy- diff(F1,x),trig));\n if has(hpx,y) then\n if prntflg then\n \+ print(hpx,`should not depend on`,y);print(``);\n end if;\n \+ return NULL;\n end if;\n if prntflg then\n print(`where ` ,Diff(h(x),x)=Mxy-Diff(F1,x));\n print(` =`,hpx);print(` `);\n end if;\n intg := Int(hpx,x);\n val := value(intg);\n if indets(val,'specfunc(anything,int)')=\{\} then\n hx := simplify( combine(val,trig));\n else\n if prntflg then\n print(`c annot find `,intg);print(``);\n end if;\n return NULL;\n e nd if;\n if prntflg then\n print(`so `,h(x)=intg);\n print (` = `,hx);print(``);\n end if;\n Fxy := removeconsts(F1+hx);\n if prntflg then\n print(F(x,y)=Fxy,` satisfies . .`);\n \+ print(Diff(F(x,y),x)=Mxy);\n print(Diff(F(x,y),y)=Nxy);print(``); \n end if;\n soln := Fxy=C[j1];\n\n 1111:\n if initcond then\n if prntflg then\n print(subs(y=yx,soln));print(``);\n \+ end if;\n assignedconst := false;\n const := traperror(ev al(subs(\{x=x0,y=y0\},Fxy)));\n if const<>lasterror then\n \+ soln := simplify(subs(C[j1]=const,soln));\n assignedconst := true;\n end if;\n end if;\n\n # Try to obtain y explicitly. \n sols := [];\n if frmt='explicit' then\n sols := goodsols([ solve(soln,y)]);\n end if;\n\n if initcond then\n if nops(sol s)=0 and not assignedconst then\n error \"impossible initial c ondition\"\n end if;\n # Check which solution fits the initi al condition.\n if assignedconst then\n sol := []; \n \+ for i from 1 to nops(sols) do\n yy0 := traperror(si mplify(value(eval(subs(x=x0,sols[i])))));\n if yy0<>lasterr or then\n sgny0 := signum(0,y0-yy0,0);\n i f has(sgny0,signum) and type(y0,realcons) then\n sgny 0 := signum(0,fnormal(evalf(y0-yy0,Digits+2)),0);\n end \+ if;\n if sgny0=0 then\n sol := [simplif y(sols[i])];\n break\n end if;\n \+ end if;\n end do;\n sols := sol;\n else # Ha ve another go at finding the constant.\n for i from 1 to nops( sols) do\n cc := traperror([solve(y0=subs(x=x0,sols[i]),C[j 1])]);\n if cc=lasterror then\n error \"impos sible initial condition\"\n end if;\n if cc<>[] \+ then\n const := cc[1];\n sols := [simplify (subs(C[j1]=const,sols[i]))];\n break;\n end \+ if;\n end do;\n if cc=[] then\n error \"imp ossible initial condition\"\n end if;\n end if;\n if prntflg then\n print(`Applying the initial condition . . \+ `,\n C[j1]=simplify(const));print( ``);\n end if;\n end if;\n\n if nops(sols)>0 then\n retu rn seq(yx=simplify(sols[i]),i=1..nops(sols));\n else\n return s ubs(y=yx,soln);\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Examples are given in the following sections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 " desolveEX" }{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 3 " " }{XPPEDIT 18 0 "-1+y*e xp(x*y)+y*cos(x*y)+``(1+x*exp(x*y)+x*cos(x*y));" "6#,*\"\"\"!\"\"*&%\" yGF$-%$expG6#*&%\"xGF$F'F$F$F$*&F'F$-%$cosG6#*&F,F$F'F$F$F$-%!G6#,(F$F $*&F,F$-F)6#*&F,F$F'F$F$F$*&F,F$-F/6#*&F,F$F'F$F$F$F$" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "M := (x,y)->-1+y*exp(x*y)+y*cos(x*y):\nN := (x,y)->1+x*exp(x*y)+x *cos(x*y):\nde := M(x,y(x))+N(x,y(x))*diff(y(x),x)=0:\ndesolveEX(de,in fo=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Test~for~exactness~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(!\"\"\"\"\"*&%\"yGF)-%$expG6#*&%\"xGF)F+F)F)F)*&F+F )-%$cosGF.F)F)F+,*F,F)*(F+F)F0F)F,F)F)F2F)*(F+F)-%$sinGF.F)F0F)F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(\"\"\"F(*&%\"xGF(-%$expG6 #*&F*F(%\"yGF(F(F(*&F*F(-%$cosGF-F(F(F*,*F+F(*(F/F(F*F(F+F(F(F1F(*(F/F (-%$sinGF-F(F*F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.Exact~DE~.~.~G/,*!\"\"\"\"\"*&%\"yGF'-%$ex pG6#*&%\"xGF'F)F'F'F'*&F)F'-%$cosGF,F'F'*&,(F'F'*&F.F'F*F'F'*&F.F'F0F' F'F'-%!G6#*&%#dyGF'%#dxGF&F'F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%Let~G/-%\"FG6$%\"xG%\"yG,&-% $IntG6$,(!\"\"\"\"\"*&F)F0-%$expG6#*&F(F0F)F0F0F0*&F)F0-%$cosGF4F0F0F( F0-%\"gG6#F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G,*%\"xG!\" \"-%$expG6#*&F%\"\"\"%\"yGF+F+-%$sinGF)F+-%\"gG6#F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'where~G/- %%DiffG6$-%\"gG6#%\"yGF+,*\"\"\"F-*&%\"xGF--%$expG6#*&F/F-F+F-F-F-*&F/ F--%$cosGF2F-F--F&6$,(F/!\"\"F0F--%$sinGF2F-F+F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,~~~~~~~~~~=G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$so~G/-%\"gG6#%\"yG-%$IntG6$ \"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G%\"yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\" FG6$%\"xG%\"yG,*F'!\"\"-%$expG6#*&F'\"\"\"F(F/F/-%$sinGF-F/F(F/%0~~sat isfies~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"FG6$%\"x G%\"yGF*,(!\"\"\"\"\"*&F+F.-%$expG6#*&F*F.F+F.F.F.*&F+F.-%$cosGF2F.F. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF+,( \"\"\"F-*&F*F--%$expG6#*&F*F-F+F-F-F-*&F*F--%$cosGF1F-F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*%\"xG! \"\"-%$expG6#*&F%\"\"\"-%\"yG6#F%F+F+-%$sinGF)F+F,F+&%\"CG6#F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dsolve(de,implicit);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,%\"xG! \"\"-%$expG6#*&F%\"\"\"-%\"yG6#F%F+F+-%$sinGF)F+F,F+%$_C1GF+\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1A" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "-1+y*exp(x*y)+y*cos(x*y)+ ``(1+x*exp(x*y)+x*cos(x*y));" "6#,*\"\"\"!\"\"*&%\"yGF$-%$expG6#*&%\"x GF$F'F$F$F$*&F'F$-%$cosG6#*&F,F$F'F$F$F$-%!G6#,(F$F$*&F,F$-F)6#*&F,F$F 'F$F$F$*&F,F$-F/6#*&F,F$F'F$F$F$F$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy /dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "M := (x,y)->-1+y* exp(x*y)+y*cos(x*y):\nN := (x,y)->1+x*exp(x*y)+x*cos(x*y):\nde := M(x, y(x))+N(x,y(x))*diff(y(x),x)=0:\ndesolveEX(de,scheme=alt,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Test~for~exactness~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%Di ffG6$,(!\"\"\"\"\"*&%\"yGF)-%$expG6#*&%\"xGF)F+F)F)F)*&F+F)-%$cosGF.F) F)F+,*F,F)*(F+F)F0F)F,F)F)F2F)*(F+F)-%$sinGF.F)F0F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(\"\"\"F(*&%\"xGF(-%$expG6#*&F*F(%\"yG F(F(F(*&F*F(-%$cosGF-F(F(F*,*F+F(*(F/F(F*F(F+F(F(F1F(*(F/F(-%$sinGF-F( F*F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.Exact~DE~.~.~G/,*!\"\"\"\"\"*&%\"yGF'-%$expG6#*&%\"xG F'F)F'F'F'*&F)F'-%$cosGF,F'F'*&,(F'F'*&F.F'F*F'F'*&F.F'F0F'F'F'-%!G6#* &%#dyGF'%#dxGF&F'F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%%Let~G/-%\"FG6$%\"xG%\"yG,&-%$IntG6$, (\"\"\"F/*&F(F/-%$expG6#*&F(F/F)F/F/F/*&F(F/-%$cosGF3F/F/F)F/-%\"hG6#F (F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G,*%\"yG\"\"\"-%$expG6# *&%\"xGF&F%F&F&-%$sinGF)F&-%\"hG6#F+F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'where~G/-%%DiffG6$-%\"h G6#%\"xGF+,*!\"\"\"\"\"*&%\"yGF.-%$expG6#*&F+F.F0F.F.F.*&F0F.-%$cosGF3 F.F.-F&6$,(F0F.F1F.-%$sinGF3F.F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %,~~~~~~~~~~=G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$so~G/-%\"hG6#%\"xG-%$IntG6$!\"\"F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G,$%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"FG6$% \"xG%\"yG,*F'!\"\"-%$expG6#*&F'\"\"\"F(F/F/-%$sinGF-F/F(F/%0~~satisfie s~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"FG6$%\"xG%\"y GF*,(!\"\"\"\"\"*&F+F.-%$expG6#*&F*F.F+F.F.F.*&F+F.-%$cosGF2F.F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF+,(\"\" \"F-*&F*F--%$expG6#*&F*F-F+F-F-F-*&F*F--%$cosGF1F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*%\"xG!\"\" -%$expG6#*&F%\"\"\"-%\"yG6#F%F+F+-%$sinGF)F+F,F+&%\"CG6#F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ds olve(de,implicit);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,%\"xG!\"\"-%$ expG6#*&F%\"\"\"-%\"yG6#F%F+F+-%$sinGF)F+F,F+%$_C1GF+\"\"!" }}}{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 3 " " }{XPPEDIT 18 0 "cos(x)*sin(x)-x*y^2+``(y*(1-x^2)); " "6#,(*&-%$cosG6#%\"xG\"\"\"-%$sinG6#F(F)F)*&F(F)*$%\"yG\"\"#F)!\"\"- %!G6#*&F/F),&F)F)*$F(F0F1F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "M := (x,y)->cos(x)*sin( x)-x*y^2:\nN := (x,y)->y*(1-x^2):\nde := M(x,y(x))+N(x,y(x))*diff(y(x) ,x)=0:\ndesolveEX(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Test~for~exactness~.~.G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,&*&-%$cosG6#%\"xG\"\"\"-%$ sinGF+F-F-*&F,F-)%\"yG\"\"#F-!\"\"F2,$*&F,F-F2F-!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&%\"yG\"\"\",&F)F)*$)%\"xG\"\"#F)!\"\"F) F-,$*&F-F)F(F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.Exact~DE~.~.~G/,(*&-%$cosG6#%\"xG\"\"\"-%$sin GF)F+F+*&F*F+)%\"yG\"\"#F+!\"\"*(F0F+,&F+F+*$)F*F1F+F2F+-%!G6#*&%#dyGF +%#dxGF2F+F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%%Let~G/-%\"FG6$%\"xG%\"yG,&-%$IntG6$,&*&-%$cosG 6#F(\"\"\"-%$sinGF2F3F3*&F(F3)F)\"\"#F3!\"\"F(F3-%\"gG6#F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G,*-%$cosG6#,$%\"xG\"\"##!\"\"\"\"%# \"\"\"F-F,*&#F/F*F/*&)F)F*F/)%\"yGF*F/F/F,-%\"gG6#F5F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'where~G/- %%DiffG6$-%\"gG6#%\"yGF+,&*&F+\"\"\",&F.F.*$)%\"xG\"\"#F.!\"\"F.F.-F&6 $,(-%$cosG6#,$F2F3#F4\"\"%#F.F=F4*&#F.F3F.*&F1F.)F+F3F.F.F4F+F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%,~~~~~~~~~~=G%\"yG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$so~G/-%\"gG6 #%\"yG-%$IntG6$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G,$*$)% \"yG\"\"#\"\"\"#F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"FG6$%\"xG%\"yG,(-%$cosG6#,$F'\"\"##!\" \"\"\"%*&#\"\"\"F.F4*&)F'F.F4)F(F.F4F4F0*&#F4F.F4F7F4F4%0~~satisfies~. ~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF* ,&*&-%$cosG6#F*\"\"\"-%$sinGF0F1F1*&F*F1)F+\"\"#F1!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF+*&F+\"\"\",&F-F-* $)F*\"\"#F-!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG,$*(,&!\"\"\"\"\"*$)F'\"\"#F,F,F+ F/#F,F/,$*&F*F,,&-%$cosG6#,$F'F/F,*&\"\"%F,&%\"CG6#F,F,F,F,F+F0F0/F$,$ F)#F+F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "dsolve(de,implicit);\ndsolve(de);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,&*$)-%\"yG6#%\"xG\"\"#\"\"\"F,*&,&*$)-%$cosGF)F+F,! \"\"%$_C1GF,F,,&F3F,*$)F*F+F,F,F3F3\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*&,&!\"\"\"\"\"*$)F'\"\"#F+F+F*,$*&F)F+,&*$)-%$cos GF&F.F+F+%$_C1GF*F+F*#F+F./F$,$F(F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2A" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "cos(x)*sin(x)-x*y^2+``(y*(1-x^2));" "6#,(*&-%$cosG6#%\" xG\"\"\"-%$sinG6#F(F)F)*&F(F)*$%\"yG\"\"#F)!\"\"-%!G6#*&F/F),&F)F)*$F( F0F1F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\" \"%#dxG!\"\"\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "M := (x,y)->cos(x)*sin(x)-x*y^2:\nN := (x,y) ->y*(1-x^2):\nde := M(x,y(x))+N(x,y(x))*diff(y(x),x)=0:\ndesolveEX(de, scheme=alt,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%7Test~for~exactness~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,&*&-%$cosG6#%\"xG\"\"\"-%$sinGF+F-F -*&F,F-)%\"yG\"\"#F-!\"\"F2,$*&F,F-F2F-!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&%\"yG\"\"\",&F)F)*$)%\"xG\"\"#F)!\"\"F)F-, $*&F-F)F(F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.Exact~DE~.~.~G/,(*&-%$cosG6#%\"xG\"\"\"-%$sinGF) F+F+*&F*F+)%\"yG\"\"#F+!\"\"*(F0F+,&F+F+*$)F*F1F+F2F+-%!G6#*&%#dyGF+%# dxGF2F+F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%Let~G/-%\"FG6$%\"xG%\"yG,&*&,&!\"\"\"\"\"*$)F(\" \"#F.F.F.-%$IntG6$F)F)F.F--%\"hG6#F(F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G,(*$)%\"yG\"\"#\"\"\"#F)F(*&#F)F(F)*&)%\"xGF(F)F&F)F)! \"\"-%\"hG6#F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%'where~G/-%%DiffG6$-%\"hG6#%\"xGF+,(*&-%$cosGF* \"\"\"-%$sinGF*F0F0*&F+F0)%\"yG\"\"#F0!\"\"-F&6$,&*$F4F0#F0F6*&#F0F6F0 *&)F+F6F0F4F0F0F7F+F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,~~~~~~~~~~= G,$-%$sinG6#,$%\"xG\"\"##\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$so~G/-%\"hG6#%\"xG-%$IntG6$,$- %$sinG6#,$F(\"\"##\"\"\"F1F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~ ~~G,$-%$cosG6#,$%\"xG\"\"##!\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"FG6$%\"xG%\"yG,(-%$cosG 6#,$F'\"\"##!\"\"\"\"%*&#\"\"\"F.F4*&)F'F.F4)F(F.F4F4F0*&#F4F.F4F7F4F4 %0~~satisfies~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"F G6$%\"xG%\"yGF*,&*&-%$cosG6#F*\"\"\"-%$sinGF0F1F1*&F*F1)F+\"\"#F1!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF+*&F +\"\"\",&F-F-*$)F*\"\"#F-!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG,$*(,&!\"\"\"\"\"*$ )F'\"\"#F,F,F+F/#F,F/,$*&F*F,,&-%$cosG6#,$F'F/F,*&\"\"%F,&%\"CG6#F,F,F ,F,F+F0F0/F$,$F)#F+F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "dsolve(de,implicit);\ndsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)-%\"yG6#%\"xG\"\"#\"\"\"F,*&,& *$)-%$cosGF)F+F,!\"\"%$_C1GF,F,,&F3F,*$)F*F+F,F,F3F3\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*&,&!\"\"\"\"\"*$)F'\"\"#F+F+F*,$ *&F)F+,&*$)-%$cosGF&F.F+F+%$_C1GF*F+F*#F+F./F$,$F(F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "cos(x)*sin(x)-x*y^2+``(y*(1-x^2));" " 6#,(*&-%$cosG6#%\"xG\"\"\"-%$sinG6#F(F)F)*&F(F)*$%\"yG\"\"#F)!\"\"-%!G 6#*&F/F),&F)F)*$F(F0F1F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0; " "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "M := (x,y)->3*x^2*sin(y)^2: \nN := (x,y)->2*x^3*sin(y)*cos(y)-2*exp(2*y):\nde := M(x,y(x))+N(x,y(x ))*diff(y(x),x)=0:\ndesolveEX(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Test~for~exact ness~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,$*&)%\"xG\"\" #\"\"\")-%$sinG6#%\"yGF+F,\"\"$F1,$*(F)F,F.F,-%$cosGF0F,\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,&*()%\"xG\"\"$\"\"\"-%$sin G6#%\"yGF,-%$cosGF/F,\"\"#*&F3F,-%$expG6#,$F0F3F,!\"\"F*,$*()F*F3F,F-F ,F1F,\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.Exact~DE~.~.~G/,&*&)%\"xG\"\"#\"\"\")-%$sinG6#%\"yGF) F*\"\"$*&,&*()F(F0F*F,F*-%$cosGF.F*F)*&F)F*-%$expG6#,$F/F)F*!\"\"F*-%! G6#*&%#dyGF*%#dxGF " 0 "" {MPLTEXT 1 0 20 "dsolve(de,implicit);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**$)%\"x G\"\"$\"\"\"#F)\"\"#*&#F)F+F)*&F&F)-%$cosG6#,$-%\"yG6#F'F+F)F)!\"\"-%$ expGF1F6%$_C1GF)\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolveEX" } {TEXT -1 30 ": particular solution examples" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 34 "We solve the initial value problem" } }{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "-1+y*exp(x*y)+y*co s(x*y)+``(1+x*exp(x*y)+x*cos(x*y));" "6#,*\"\"\"!\"\"*&%\"yGF$-%$expG6 #*&%\"xGF$F'F$F$F$*&F'F$-%$cosG6#*&F,F$F'F$F$F$-%!G6#,(F$F$*&F,F$-F)6# *&F,F$F'F$F$F$*&F,F$-F/6#*&F,F$F'F$F$F$F$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 5 ", \+ " }{XPPEDIT 18 0 "y(0) = 4/3" "6#/-%\"yG6#\"\"!*&\"\"%\"\"\"\"\"$!\"\" " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 35 "and plot the graph o f the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 164 "M := (x,y)->-1+y*exp(x*y)+y*cos(x*y):\nN := ( x,y)->1+x*exp(x*y)+x*cos(x*y):\nde := M(x,y(x))+N(x,y(x))*diff(y(x),x) =0:\nic := y(0)=4/3;\nsol := desolveEX(\{de,ic\},y(x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!#\"\"%\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/,*%\"xG!\"\"-%$expG6#*&F'\"\"\"-%\"yG6#F'F -F--%$sinGF+F-F.F-#\"\"(\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "eq := subs(y(x)=y,sol);\npl ots[implicitplot](eq,x=-4..4,y=-4..4,grid=[100,100],\n thickness=2, color=coral);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,*%\"xG!\"\"-% $expG6#*&F'\"\"\"%\"yGF-F--%$sinGF+F-F.F-#\"\"(\"\"$" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6&-%'CURVESG6bal7$7$$!3?jkkkkkkC !#<$\"3oB[geJ8oZ!#=7$$!3+1Vk#=2!3DF*$\"3!3GDDDDDD&F-7$F.7$$!3k\"*[[qx \\/DF*$\"3)=c44JQ5l&F-7$7$$!3+t9+:Ep1DF*$\"3_)3111111'F-F47$F:7$$!3WVB .[Zb([#F*$\"3)4*[Y%*))o*G'F-7$7$F($\"3kn4;#oU]p'F-F@7$7$$!3I#QQQQQQQ#F *$\"3oW'=wepT?%F-7$$!3R$o,TlBEiVX#F*$\"3E'*oooooooF-FV7$7$Fgn$\" 3;&*oooooooF-7$$!3qv%[NVodT#F*$\"3/Hyylt)z=(F-7$7$FK$\"3aOcvA(*H_vF-F_ o7$FJ7$$!3d)\\S(=8#zL#F*$\"3;NcY$zt_)RF-7$7$$!3V,......BF*$\"3^:EuAy0< QF-Fio7$Feo7$$!3Eo#*z?JOxBF*$\"3cklPY],7wF-7$7$$!3[DV\"Q8qBP#F*$\"3+/x wwwwwwF-Fep7$F[q7$$!3^`NEO^uJBF*$\"3![A+\"4g\"R'zF-7$7$F`p$\"325Q+._BN #)F-Faq7$7$$!3a?AAAAAAAF*$\"3!*\\%*zx\\b&\\$F-7$$!3)zLu85?RE#F*$\"3AkO OOOOOOF-7$F`rF_p7$7$F`p$\"3'*3Q+._BN#)F-7$$!3/D;hfO())G#F*$\"3))[\"\\NVAF*$\"3Kcor\"Qvhp)F-7$7$F\\r$\"3c1mP5.`z))F-Fis7$F[r7$$! 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ut$F37$$\"3/+++X/%\\!**F*$\"3?cxZ5^&e)RF37$$\"3\"*******z._=**F*$\"3% \\`**y0H[H%F37$$\"3@+++?.5K**F*$\"3#H0@^VrLp%F37$$\"33+++b-oX**F*$\"33 \">xq8FYB&F37$$\"3'********=g#f**F*$\"3ly.n'fO(HgF37$$\"3C+++I,%G(**F* $\"3U " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example \+ 3" }}{PARA 0 "" 0 "" {TEXT -1 34 "We solve the initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "2*x/y-x^2/(y^2); " "6#,&*(\"\"#\"\"\"%\"xGF&%\"yG!\"\"F&*&F'F%*$F(F%F)F)" }{XPPEDIT 18 0 "``(dy/dx) = 0;" "6#/-%!G6#*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "y(1) = 1" "6#/-%\"yG6#\"\"\"F'" }{TEXT -1 1 " ." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "de := 2*x/y(x)-x^2/y(x)^2*diff(y(x),x)=0;\nic := y(1) =1;\ndesolveEX(\{de,ic\},info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#deG/,&*(\"\"#\"\"\"%\"xGF)-%\"yG6#F*!\"\"F)*(F*F(F+!\"#-%%diffG6$ F+F*F)F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\" F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Test~for~exactness~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,$*(\"\"#\"\"\"%\"xGF*%\"yG!\"\"F*F,,$*(F)F*F+F*F,!\" #F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,$*&%\"xG\"\"#%\"yG! \"#!\"\"F),$*(F*\"\"\"F)F0F+F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.Exact~DE~.~.~G/,&*(\"\"#\"\"\"% \"xGF(%\"yG!\"\"F(*(F)F'F*!\"#-%!G6#*&%#dyGF(%#dxGF+F(F+\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%%Let~G/-%\"FG6$%\"xG%\"yG,&,$-%$IntG6$F(F(*&\"\"#\"\"\"F)!\"\"F1-%\" gG6#F)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G,&*&%\"xG\"\"#%\" yG!\"\"\"\"\"-%\"gG6#F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%'where~G/-%%DiffG6$-%\"gG6#%\"yGF+,&* &%\"xG\"\"#F+!\"#!\"\"-F&6$*&F.F/F+F1F+F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,~~~~~~~~~~=G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$so~G/-%\"gG6#%\"yG-%$IntG6$\" \"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"FG6$ %\"xG%\"yG*&F'\"\"#F(!\"\"%0~~satisfies~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF*,$*(\"\"#\"\"\"F*F/F+!\" \"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF +,$*&F*\"\"#F+!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"xG\"\"#-%\"yG6#F%!\"\"&%\"CG6#\" \"\"" }}{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#/-%\"yG6#%\"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 4" }}{PARA 0 "" 0 "" {TEXT -1 34 "We solve \+ the initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "3*x^2/(y^2)-2*x^3/(y^3);" "6#,&*(\"\"$\"\"\"*$%\"xG\"\" #F&*$%\"yGF)!\"\"F&*(F)F&*$F(F%F&*$F+F%F,F," }{XPPEDIT 18 0 "``(dy/dx) = sin(x);" "6#/-%!G6#*&%#dyG\"\"\"%#dxG!\"\"-%$sinG6#%\"xG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 1" "6#/-%\"yG6#\"\"\"F'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 36 "and plot the graph of the solutio n. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "de := 3*x^2/y(x)^2-2*x^3/y(x)^3*diff(y(x),x)=sin(x); \nic := y(1)=1;\ndesolveEX(\{de,ic\},info=true);\ng := unapply(rhs(%), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&*&%\"xG\"\"#-%\"yG6#F( !\"#\"\"$**F)\"\"\"F(F.F*!\"$-%%diffG6$F*F(F0!\"\"-%$sinGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Test~for~exact ness~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,&*&%\"xG\"\"# %\"yG!\"#\"\"$-%$sinG6#F)!\"\"F+,$*&F)F*F+!\"$!\"'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%%DiffG6$,$*&%\"xG\"\"$%\"yG!\"$!\"#F),$*&F)\"\"#F+ F,!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.Exact~DE~.~.~G/,(*&%\"xG\"\"#%\"yG!\"#\"\"$-%$sinG6#F '!\"\"**F(\"\"\"F'F+F)!\"$-%!G6#*&%#dyGF1%#dxGF/F1F/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%Let~G/ -%\"FG6$%\"xG%\"yG,&*&F)!\"#-%$IntG6$,&*$)F(\"\"#\"\"\"!\"$*&-%$sinG6# F(F4)F)F3F4F4F(F4!\"\"-%\"gG6#F)F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %'~~=~~~G,&*&,&*$)%\"xG\"\"$\"\"\"F+*&-%$cosG6#F)F+)%\"yG\"\"#F+F+F+F1 !\"#F+-%\"gG6#F1F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'where~G/-%%DiffG6$-%\"gG6#%\"yGF+,&*&%\"xG\" \"$F+!\"$!\"#-F&6$*&,&*$)F.F/\"\"\"F8*&-%$cosG6#F.F8)F+\"\"#F8F8F8F+F1 F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,~~~~~~~~~~=G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%$so~G/-%\"gG6#%\"yG-%$IntG6$\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$/-%\"FG6$%\"xG%\"yG*&,&*$)F'\"\"$\"\"\"F.*&-%$co sG6#F'F.)F(\"\"#F.F.F.F(!\"#%0~~satisfies~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF*,&*&F*\"\"#F+!\"#\"\"$-% $sinG6#F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"FG6$% \"xG%\"yGF+,$*&F*\"\"$F+!\"$!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&*$)%\"xG\"\"$\"\"\"F**&-%$co sG6#F(F*)-%\"yGF.\"\"#F*F*F*F0!\"#&%\"CG6#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~i nitial~condition~.~.~~G/&%\"CG6#\"\"\",&F(F(-%$cosGF'F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6# %\"xG,$*(,(-%$cosGF&\"\"\"F-!\"\"-F,6#F-F.F.,$*&F*F-F'F-F.#F-\"\"#F'F- F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG% &arrowGF(,$*(,(-%$cosG6#9$\"\"\"F3!\"\"-F06#F3F4F4,$*&F.F3F2F3F4#F3\" \"#F2F3F4F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "plot(g(x),x=0..15);" }}{PARA 13 "" 1 "" {GLPLOT2D 325 211 211 {PLOTDATA 2 "6%-%'CURVESG6$7eo7$$\"\"!F)F(7$$\"3 ')*****\\7t&pK!#=$\"36A7V@)4sU#F-7$$\"3$****\\(=7T9hF-$\"3(zmh(e!\\)Gc F-7$$\"3X****\\(=HPJ*F-$\"3%p%4%*e\"F=7$$\"3%)**\\(=276(=F=$\"3/] ;%REb)))=F=7$$\"3'***\\(o**3)y@F=$\"3%\\u!HjIC8AF=7$$\"3/+](ofHq\\#F=$ \"3C=A^dxjzDF=7$$\"3.+]Pf'HU\"GF=$\"3q'p?WzMN*HF=7$$\"33++]7*309$F=$\" 3s0)H&F=$\"3ut&*yk;9F7Ffp7$$\"3Y**\\(=-p 6j&F=$\"3kl9*zV;xa\"Ffp7$$\"3d*****\\2Mg#fF=$\"3at#RgbEr&=Ffp7$$\"3%)* \\PM#4z(3'F=$\"3R<*Q!*eA%3?Ffp7$$\"35+](=xZ&\\iF=$\"3Nl`\"4vbV7#Ffp7$$ \"37+D\"y:jlK'F=$\"37g(oOf&*H;#Ffp7$$\"38++vV&yNS'F=$\"3+8trXz!**=#Ffp 7$$\"38](=nB'3UkF=$\"3)\\4!\\/[&*)>#Ffp7$$\"39+voHRf!['F=$\"38bmOk\\80 AFfp7$$\"3:]ilA;5>lF=$\"3[GpZpxa3AFfp7$$\"3;+]i:$4wb'F=$\"3*4@B*eLK4AF fp7$$\"3Y\\iS\"zmzj'F=$\"3*>#))yWi:.AFfp7$$\"3k*\\(=nUK=nF=$\"3kS#)*H! )oy=#Ffp7$$\"3#)\\(oHu\"o)z'F=$\"3?D$GV5'Hl@Ffp7$$\"3-++v=#R!zoF=$\"3* R;x32ns8#Ffp7$$\"3q+]P4A@urF=$\"3t`3F,,^7?Ffp7$$\"3I++Dchf#\\(F=$\"3=& f9ENhE)=Ffp7$$\"3))**\\(of2L#yF=$\"3Qe9VRD%4y\"Ffp7$$\"3M**\\7yG>6\")F =$\"3kv3*f6zVs\"Ffp7$$\"3w++voo6A%)F=$\"3M+,ykfR&p\"Ffp7$$\"3q*****\\x JLu)F=$\"31uz1G:Y)p\"Ffp7$$\"3W++v$*ydd!*F=$\"3O@e[q_?LFfp 7$$\"3*)****\\2mD+5Ffp$\"3unor#3i90#Ffp7$$\"3*****\\i0XE.\"Ffp$\"3#eLE +)*fvD#Ffp7$$\"3%**\\(o/Q*>1\"Ffp$\"3\\Lm/&oVg]#Ffp7$$\"3=++vQ(zS4\"Ff p$\"3%)f0LM`SlGFfp7$$\"33](o/#\\<46Ffp$\"3'4qV6!fztIFfp7$$\"3***\\(=-, FC6Ffp$\"3+\\1u$\\Unl4@&f$Ffp7$$\"33+ v$4tFe:\"Ffp$\"3iH)egduh\"RFfp7$$\"3)*\\P41WDr6Ffp$\"3ke$*HQ2GlUFfp7$$ \"3!****\\73\"o'=\"Ffp$\"3%*)pEt`@Ik%Ffp7$$\"3'*\\(o/QJG?\"Ffp$\"3Qvv< GJ$H0&Ffp7$$\"3-+voz;)*=7Ffp$\"3o*pV\"epaZaFfp7$$\"3/]PMPj`M7Ffp$\"3-3 CuAjlsdFfp7$$\"31+++&*44]7Ffp$\"3dv@R_k:,gFfp7$$\"3)\\i:grnSD\"Ffp$\"3 7]R#e\"*4*RgFfp7$$\"31]7.PW/e7Ffp$\"3[3Z7!4f*pgFfp7$$\"3;vo/e6-i7Ffp$ \"3#eD1@1f64'Ffp7$$\"33+D1zy*fE\"Ffp$\"3)ff_gE&\\.hFfp7$$\"3+D\"y+gu*p 7Ffp$\"3Gs!R/;#32hFfp7$$\"35]P4@8&RF\"Ffp$\"36T'e4$p:-hFfp7$$\"3?v$4@/ GzF\"Ffp$\"3]V%3Kvo!*3'Ffp7$$\"35+]7jZ!>G\"Ffp$\"3Sn'zkWk#ogFfp7$$\"35 ]i:I*zwH\"Ffp$\"3&[f/)om&3#fFfp7$$\"34+v=(4bMJ\"Ffp$\"37,o\">;2+q&Ffp7 $$\"3;++]xlWU8Ffp$\"3@2z,VVEC_Ffp7$$\"39+]i&3ucP\"Ffp$\"3k1\"3f&G?>ZFf p7$$\"3\"******\\;$R09Ffp$\"3uz*f5<#ekVFfp7$$\"38+v=-*zqV\"Ffp$\"3%**3 )fD " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 34 "We solve the i nitial value problem" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2*(6*y+1)+``(2*x^3+3*y^2)*``(dy/dx) = 0;" "6#/,&*&%\"xG\"\"#,& *&\"\"'\"\"\"%\"yGF+F+F+F+F+F+*&-%!G6#,&*&F'F+*$F&\"\"$F+F+*&F4F+*$F,F 'F+F+F+-F/6#*&%#dyGF+%#dxG!\"\"F+F+\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 4 ", " }}{PARA 0 " " 0 "" {TEXT -1 66 "and plot the graph of the solution along with the \+ gradient field. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "de := x^2*(6*y(x)+1)+(2*x^3+3*y(x)^2)*diff (y(x),x)=0;\nic := y(0)=1;\ndesolveEX(\{de,ic\},info=true);\ng := unap ply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&*&)%\"xG\" \"#\"\"\",&*&\"\"'F+-%\"yG6#F)F+F+F+F+F+F+*&,&*&F*F+)F)\"\"$F+F+*&F6F+ )F/F*F+F+F+-%%diffG6$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#%7Test~for~exactness~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&)%\"xG\"\"#\"\"\",&*&\"\"'F+%\"yGF +F+F+F+F+F/,$*&F.F+F(F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG 6$,&*&\"\"#\"\"\")%\"xG\"\"$F*F**&F-F*)%\"yGF)F*F*F,,$*&\"\"'F*)F,F)F* F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.Exact~DE~.~.~G/,&*&)%\"xG\"\"#\"\"\",&*&\"\"'F*%\"yGF *F*F*F*F*F**&,&*&F)F*)F(\"\"$F*F**&F3F*)F.F)F*F*F*-%!G6#*&%#dyGF*%#dxG !\"\"F*F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%Let~G/-%\"FG6$%\"xG%\"yG,&*&,&*&\"\"'\"\"\"F)F/F /F/F/F/-%$IntG6$*$)F(\"\"#F/F(F/F/-%\"gG6#F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G,(*(\"\"#\"\"\")%\"xG\"\"$F'%\"yGF'F'*&F*!\"\" F)F*F'-%\"gG6#F+F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'where~G/-%%DiffG6$-%\"gG6#%\"yGF+,(*&\"\"#\" \"\")%\"xG\"\"$F/F/*&F2F/)F+F.F/F/-F&6$,&*(F.F/F0F/F+F/F/*&#F/F2F/*$F0 F/F/F/F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,~~~~~~~~~~=G,$*&\" \"$\"\"\")%\"yG\"\"#F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%$so~G/-%\"gG6#%\"yG-%$IntG6$,$*&\"\"$ \"\"\")F(\"\"#F/F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G*$)% \"yG\"\"$\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"FG6$%\"xG%\"yG,(*(\"\"#\"\"\")F'\"\"$F,F(F,F, *&#F,F.F,*$F-F,F,F,*$)F(F.F,F,%0~~satisfies~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 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}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 44 "Determine whether the differential equati on " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2*x-1+``(3*y+ 7);" "6#,(*&\"\"#\"\"\"%\"xGF&F&F&!\"\"-%!G6#,&*&\"\"$F&%\"yGF&F&\"\"( F&F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#d xG!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "is exact, and if so find its general solution." }}{PARA 0 "" 0 "" {TEXT -1 37 " _____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "__ ___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }} {PARA 0 "" 0 "" {TEXT -1 44 "Determine whether the differential equati on " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "4*x^3+4*x*y+` `(2*x^2+2*y-1);" "6#,(*&\"\"%\"\"\"*$%\"xG\"\"$F&F&*(F%F&F(F&%\"yGF&F& -%!G6#,(*&\"\"#F&*$F(F1F&F&*&F1F&F+F&F&F&!\"\"F&" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "is exact, and if so find its \+ general solution." }}{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 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 52 "_______________________________ _____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 44 "Determine whether the differential equation " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y^3-y^2*sin(x)-x+``(3*x*y^2+2*y*cos (x));" "6#,**$%\"yG\"\"$\"\"\"*&F%\"\"#-%$sinG6#%\"xGF'!\"\"F-F.-%!G6# ,&*(F&F'F-F'F%F)F'*(F)F'F%F'-%$cosG6#F-F'F'F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "is exact, and if so find its \+ general solution." }}{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 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 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 44 "Determ ine whether the differential equation " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y*(1-2*x^2)*exp(-x^2-y^2)+``(x*(1-2*y^2)*exp(-x ^2-y^2));" "6#,&*(%\"yG\"\"\",&F&F&*&\"\"#F&*$%\"xGF)F&!\"\"F&-%$expG6 #,&*$F+F)F,*$F%F)F,F&F&-%!G6#*(F+F&,&F&F&*&F)F&*$F%F)F&F,F&-F.6#,&*$F+ F)F,*$F%F)F,F&F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG \"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 47 " is exact, and if so, find its general solution." }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }} {PARA 0 "" 0 "" {TEXT -1 32 "Solve the initial value problem " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x+y)^2+``(2*x*y+x^2- 1);" "6#,&*$,&%\"xG\"\"\"%\"yGF'\"\"#F'-%!G6#,(*(F)F'F&F'F(F'F'*$F&F)F 'F'!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0,y(1) = 1;" "6$/* &%#dyG\"\"\"%#dxG!\"\"\"\"!/-%\"yG6#F&F&" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 42 "and plot the graph of the solution (using " } {TEXT 0 12 "implicitplot" }{TEXT -1 16 " if necessary). " }}{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 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 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 32 "Solve the initial value problem \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x)+y+``(2+x+y *exp(y));" "6#,(-%$expG6#%\"xG\"\"\"%\"yGF(-%!G6#,(\"\"#F(F'F(*&F)F(-F %6#F)F(F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0,y(0) = 1;" "6$/ *&%#dyG\"\"\"%#dxG!\"\"\"\"!/-%\"yG6#F)F&" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 42 "and plot the graph of the solution (using " } {TEXT 0 12 "implicitplot" }{TEXT -1 16 " if necessary). " }}{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 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 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 32 "Solve the initial value problem \+ " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y^2*cos(x)-3*x^2 *y-2*x+``(2*y*sin(x)-x^3+ln(y));" "6#,**&%\"yG\"\"#-%$cosG6#%\"xG\"\" \"F+*(\"\"$F+*$F*F&F+F%F+!\"\"*&F&F+F*F+F/-%!G6#,(*(F&F+F%F+-%$sinG6#F *F+F+*$F*F-F/-%#lnG6#F%F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0 ,y(0) = exp(1);" "6$/*&%#dyG\"\"\"%#dxG!\"\"\"\"!/-%\"yG6#F)-%$expG6#F &" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "and plot the graph o f the solution (using " }{TEXT 0 12 "implicitplot" }{TEXT -1 16 " if n ecessary). " }}{PARA 0 "" 0 "" {TEXT -1 37 "__________________________ ___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q8 " }}{PARA 0 "" 0 "" {TEXT -1 33 "Solve the initial value problem " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "( 12*y^2-x*exp(x*y))*``(dy/dx)=y*exp(x*y), y(0)=-1" "6$/*&,&*&\"#7\"\"\" *$%\"yG\"\"#F(F(*&%\"xGF(-%$expG6#*&F-F(F*F(F(!\"\"F(-%!G6#*&%#dyGF(%# dxGF2F(*&F*F(-F/6#*&F-F(F*F(F(/-F*6#\"\"!,$F(F2" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 42 "and plot the graph of the solution (using " }{TEXT 0 12 "implicitplot" }{TEXT -1 16 " if necessary). " }}{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 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 }