{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 "" 265 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 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 "Bullet 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 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" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 46 "Homegeneous first order different ial equations" }}{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 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 52 "Solving 1st order homogeneous differential equations" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 8 "dsolveHG" }{TEXT -1 71 " in the nex t section tries to solve a differential equation of the form" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = F(y/x);" "6#/*&%#dy G\"\"\"%#dxG!\"\"-%\"FG6#*&%\"yGF&%\"xGF(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by means of the substitution " }{XPPEDIT 18 0 "y \+ = v*x;" "6#/%\"yG*&%\"vG\"\"\"%\"xGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "dy/dx = v+x;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&%\"vGF&%\"xGF&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx;" "6#*&%#dvG\"\"\"%#dxG!\"\"" }{TEXT -1 36 ", the differential equation becomes " }{XPPEDIT 18 0 "v+x;" "6#,&% \"vG\"\"\"%\"xGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx = F(v);" "6#/ *&%#dvG\"\"\"%#dxG!\"\"-%\"FG6#%\"vG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "This equation can be wr itten in the form" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " x;" "6#%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx = F(v)-v;" "6#/*&% #dvG\"\"\"%#dxG!\"\",&-%\"FG6#%\"vGF&F-F(" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 2 "or" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/(F(v)-v);" "6#*&\"\"\"F$,&-%\"FG6#%\"vGF$F)!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx = 1/x;" "6#/*&%#dvG\"\"\"%#dxG!\"\"*&F&F&% \"xGF(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "and so has sep arable variables." }}{PARA 0 "" 0 "" {TEXT -1 39 "Integrating both sid es with respect to " }{TEXT 267 1 "x" }{TEXT -1 6 " gives" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(1/(F(v)-v),v) = Int(1/ x,x);" "6#/-%$IntG6$*&\"\"\"F(,&-%\"FG6#%\"vGF(F-!\"\"F.F--F%6$*&F(F(% \"xGF.F2" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "so " }} {PARA 257 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "ln(x) = Int(1/(F(v )-v),v);" "6#/-%#lnG6#%\"xG-%$IntG6$*&\"\"\"F,,&-%\"FG6#%\"vGF,F1!\"\" F2F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 109 "If this integral can be found analytically, we shall obtain at least an implicit descr iption of the solution." }}{PARA 0 "" 0 "" {TEXT -1 35 "To suggest tha t we need to replace " }{TEXT 268 1 "v" }{TEXT -1 18 " in the result b y " }{XPPEDIT 18 0 "y(x)/x;" "6#*&-%\"yG6#%\"xG\"\"\"F'!\"\"" }{TEXT -1 52 ", we could rewrite the previous equation in the form" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "ln(x) = Int(1/(F(v)-v),v \+ = `` .. y(x)/x);" "6#/-%#lnG6#%\"xG-%$IntG6$*&\"\"\"F,,&-%\"FG6#%\"vGF ,F1!\"\"F2/F1;%!G*&-%\"yG6#F'F,F'F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 52 "In particular cases it may be possible to solve for " } {XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 12 " explicitly." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 " A procedure for solving homogeneous 1st order DE's: " }{TEXT 0 9 "deso lveHG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "desolveHG: \+ 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 75 " desolveHG( de )\n desolveHG( \{de,ic\} )\n \+ desolveHG( \{de,ic\},y(x) )" }{TEXT 262 1 "\n" }{TEXT -1 0 "" }} {PARA 256 "" 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 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 9 "desolveHG" }{TEXT -1 94 " att empts to solve a first order differential equation by making a substit ution of the form: " }{XPPEDIT 18 0 "y = _v*`.`*x;" "6#/%\"yG*(%#_vG \"\"\"%\".GF'%\"xGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "If no initial condition is given, a gen eral solution with an arbitrary constant is sought." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Options:" }{TEXT -1 1 " \n" }}{PARA 0 "" 0 "" {TEXT -1 97 "info=true/false\nThe option info=tr ue causes the intermediate steps in the solution to be printed." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "format=ex plicit/implicit" }}{PARA 0 "" 0 "" {TEXT -1 96 "When format=explicit, \+ which is the default option, an attempt is made to solve for y explici tly." }}{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 -1 154 "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 26 "desolveHG: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13190 "desolveHG := proc()\n \n loc al ff,vars,derivs,x,yx,y,df,dff,e1,intg,fv,_v,fvar,\n prntflg, Options,drv,val,rs,islog,i,j,lsic,sol,\n soln,soln2,sols,initc ond,de,ic,x0,y0,rsic,const,\n exprs,assignedconst,goodsols,frm t,pm,rightOK,lc,\n const2,separate,exp2,islg,startopts,xx,yy,e e,cc,yy0,\n sgny0,Cs,gt,jS,j1,j2,s,nvars;\n global C;\n \n # Remove any solutions involving \"RootOf\".\n goodsols := proc(s ol::list)\n local goodlist,i;\n goodlist := NULL;\n for i from 1 to nops(sol) do\n if indets(sol[i],'specfunc(anythin g,RootOf)')=\{\} then\n goodlist := goodlist,sol[i];\n \+ end if;\n end do;\n [goodlist]; \n end proc: \n \n # Separate into two factors independent and dependent of x \+ \n separate := proc(f::algebraic,x::symbol)\n \n local cansep ,opr,np,i,xops,cops,xps,cps,fx,fc;\n\n if op(0,f)<>`*` then\n \+ if member(x,indets(f,name)) then\n return [1,f]\n \+ else\n return [f,1]\n end if;\n end if;\n \n opr := [op(f)];\n np := nops(opr);\n\n # Split into \+ 2 lists of factors\n xops := NULL: cops := NULL:\n for i fro m 1 to np do\n if member(x,indets(opr[i],name)) then\n \+ xops := xops,opr[i];\n else\n cops := cops,opr[ i];\n end if;\n end do;\n xps := [xops]:\n cps \+ := [cops]:\n fx := mul(xps[i],i=1..nops(xps));\n fc := mul(c ps[i],i=1..nops(cps));\n [fc,fx];\n end proc: # of separate\n\n # Apply exp giving precedence to expressions in x.\n exp2 := proc (fx::algebraic,x::symbol)\n local sep,i,ff;\n ff := expand(f x);\n if op(0,ff)=`+` then\n return mul(exp2(op(i,ff),x), i=1..nops(ff))\n end if;\n if op(0,ff)=`*` then\n se p := separate(ff,x);\n if type(sep[2],function) and op(0,sep[2] )='ln' \n and nops(sep[2])=1 and type(op(1,sep[2]),algebr aic) then\n return op(1,sep[2])^sep[1];\n end if;\n end if;\n if type(ff,function) and op(0,ff)='ln' \n \+ and nops(ff)=1 and type(op(1,ff),algebraic) then\n return op(1,ff);\n end if;\n return simplify(exp(ff)); \n end p roc: # of exp2\n\n # Check whether fx has form of a sum of logs of e xpressions in x\n islog := proc(fx::algebraic,x::symbol)\n loca l ff,i,opf,sep,islg;\n islg := true; \n ff := expand(fx); \n if op(0,ff)=`+` then\n opf := [op(ff)];\n islg : = true;\n for i from 1 to nops(ff) do\n sep := sepa rate(opf[i],x);\n if not type(sep[2],function) or not op(0, sep[2])='ln' \n or nops(sep[2])<>1 or not type(op(1,sep[2 ]),algebraic) then\n islg := false;\n brea k;\n end if;\n end do;\n elif type(ff,function ) and op(0,ff)=`ln` and nops(ff)=1 then\n islg := evalb(type(o p(1,ff),algebraic)\n and member(x,indets(op(1,ff),name))) ;\n elif op(0,ff)=`*` then\n sep := separate(ff,x);\n \+ islg := evalb(type(sep[2],function) and op(0,sep[2])='ln' \n \+ and nops(sep[2])=1 and type(op(1,sep[2]),algebraic));\n \+ else islg := false\n end if;\n islg;\n end proc: # of islo g\n \n # start of main procedure\n if nargs>0 then \n ff := args[1]\n else\n error \"at least one argument must be supplie d\"\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) 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 na rgs>1 then\n ee := args[2];\n if type(ee,function) and nops( ee)=1 then\n yy := op(0,ee);\n xx := op(1,ee);\n \+ if type(xx,name) and type(yy,name) then\n startopts := 3 ;\n else\n error \"the 2nd argument, %1, has incorr ect form for the dependent variable\",ee;\n end if;\n end if;\n end if;\n\n prntflg := false;\n frmt := 'explicit';\n i f nargs>1 then\n Options:=[args[startopts..nargs]];\n if not type(Options,list(equation)) then\n error \"each optional arg ument must be an equation\"\n end if;\n if hasoption(Options ,'info','prntflg','Options') then \n if prntflg<>true then prn tflg := false end if;\n end if;\n if hasoption(Options,'form at','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 nops(Optio ns)>0 then\n error \"%1 is not a valid option for %2\",op(1,Op tions),procname;\n end if;\n end if;\n\n derivs := indets(de, 'specfunc(anything,diff)');\n if derivs=\{\} then\n error \"the 1st argument, %1, is invalid .. it should be a differential equation \+ or a set (or list) containing a differential equation and an initial c ondition\",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 in 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 variable in the differe ntial equation\"\n end if;\n end if;\n\n if nops(derivs)<>1 t hen\n error \"there are 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,d f)=diff and nops(df)=2 then\n yx := op(1,df);\n if not type( yx,anyfunc(name)) then\n error \"the 1st argument %1, in the d erivative, %2, is invalid .. it should be the 'unknown' dependent vari able\",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 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,nam e);\n if member(y,vars) then\n error \"%1 and %2 cannot both ap pear in the differential equation\",yx,y;\n end if;\n if op(1,yx)< >x then\n error \"the derivative, %1, does not make sense\",df;\n end if;\n\n if startopts=3 then \n if x<>xx or y<>yy then\n \+ error \"cannot solve the differential equation for %1\",ee;\n \+ end if;\n end if;\n \n if initcond then\n lsic := lhs(ic );\n if type(lsic,function) and op(0,lsic)=y and nops(lsic)=1 \n \+ and type(op(1,lsic),algebraic) then\n \+ x0 := op(1,lsic);\n if has(x0,\{x,y\}) then\n e rror \"initial condition must not involve %1 or %2\",x,y;\n en d if;\n else\n error \"initial condition is not deciphera ble\"\n end if;\n rsic := rhs(ic);\n if type(rsic,algeb raic) 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 is no t 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 el se\n error \"cannot obtain a unique expression for the derivative \"\n end if;\n\n if assigned(C) and not type(eval(C),table) then\n C := table();\n WARNING(\"C has been redefined as a table f or use as arbitrary constants\");\n end if;\n # Find the indices j 1,j2 to use in the constants\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 jS := sor t([j1,op(jS)]);\n for i to nops(jS)+1 do\n if not member(i,jS) \+ then j2 := i; break; end if; \n end do;\n \n fv := simplify(su bs(y=_v*x,e1));\n fvar := indets(fv,name);\n if member(x,fvar) or \+ member(y,fvar) then\n error \"the DE is not homogeneous\"\n end if;\n \n intg := Int(1/(fv-_v),_v);\n if prntflg then\n pri nt(``);\n print(`Homogeneous DE for which substituting `,y=_v*x ,` gives . . `);\n print(simplify(Int(1/x,x))=simplify(intg));\n print(``);\n end if;\n val := value(intg);\n if prntflg th en\n print(ln(x)=simplify(val)+C[j1]);print(``);\n end if;\n\n \+ rightOK := indets(val,'specfunc(anything,int)')=\{\};\n\n # Check \+ whether value of integral is a sum of logs.\n if frmt='explicit' and rightOK then\n islg := islog(val,_v);\n else\n islg:=fals e;\n end if;\n \n if rightOK then\n rs := subs(_v=y/x,val); \n else\n rs := Intat(1/(fv-_v),_v=y/x);\n end if;\n\n assi gnedconst := false;\n if islg then\n exprs := subs(_v=y/x,exp2( val,_v));\n soln := x=C[j2]*exprs;\n if initcond then\n \+ const2 := traperror(eval(subs(\{x=x0,y=y0\},\n \+ simplify(x/exprs))));\n if const2<>lasterror then\n \+ soln := simplify(subs(C[j2]=const2,soln));\n assi gnedconst := true;\n lc := traperror(ln(const2));\n \+ if lc<>lasterror then\n const := simplify(lc);\n \+ end if;\n end if;\n end if;\n else\n soln \+ := ln(x)=simplify(rs)+C[j1];\n if initcond then\n if righ tOK then\n const := traperror(eval(subs(\{x=x0,y=y0\},ln(x) -rs)));\n if const<>lasterror then\n soln := \+ simplify(subs(C[j1]=const,soln));\n assignedconst := tru e;\n end if;\n else\n const := ln(x0)-In tat(1/(fv-_v),_v=y0/x0);\n soln := ln(x)=int(1/(fv-_v),_v=y 0/x0..y/x)+ln(x0);\n assignedconst := true;\n end i f;\n end if;\n end if;\n\n soln2 := subs(y=yx,soln);\n\n # \+ Try to obtain y explicitly.\n sols := [];\n if frmt='explicit' and rightOK then\n sols := goodsols([solve(soln,y)]);\n end if;\n \+ \n if initcond then\n if nops(sols)=0 and not assignedconst the n\n error \"impossible initial condition\"\n end if;\n \+ # Check which solution fits the initial condition.\n if assign edconst then\n sol := []; \n for i from 1 to nops(s ols) do\n yy0 := traperror(simplify(value(eval(subs(x=x0,so ls[i])))));\n if yy0<>lasterror then\n sgny0 \+ := signum(0,y0-yy0,0);\n if has(sgny0,signum) and type(y 0,realcons) then\n sgny0 := signum(0,fnormal(evalf(y0 -yy0,Digits+2)),0);\n end if;\n if sgny0=0 then\n sol := [simplify(sols[i])];\n \+ break\n end if;\n end if;\n end do; \n sols := sol;\n else # Have another go at finding the c onstant.\n if islg then\n for i from 1 to nops(sols ) do\n cc := traperror([solve(y0=subs(x=x0,sols[i]),C[j2 ])]);\n if cc=lasterror then\n error \" impossible initial condition\"\n end if;\n \+ if cc<>[] then\n const2 := cc[1];\n \+ sols := [simplify(subs(C[j2]=const2,sols[i]))];\n br eak;\n end if;\n end do;\n if cc=[ ] or has(cc,C) then\n error \"impossible initial conditi on\"\n end if;\n else\n for i from 1 to \+ nops(sols) do\n cc := traperror([solve(y0=subs(x=x0,sols [i]),C[j1])]);\n if cc=lasterror then\n \+ error \"impossible 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=[] or has(cc,C) then\n error \"impossible initial condition\"\n end if;\n end if;\n end if;\n \+ end if;\n\n if prntflg then\n if islg then\n print(ln( x)=simplify(rs)+C[j1]);print(``);\n print(x=C[j2]*exprs);\n \+ if initcond then\n print(`Applying the initial condit ion . . `,\n C[j2]=simplify(const 2));\n end if;\n else\n print(ln(x)=simplify(rs)+ C[j1]);\n if initcond then\n print(`Applying the in itial condition . . `,\n C[j1]=si mplify(const));\n end if;\n end if;\n print(``);\n \+ end if;\n\n if nops(sols)>0 then\n return seq(yx=simplify(sols [i]),i=1..nops(sols));\n else\n return soln2;\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 "desolveHG" }{TEXT -1 27 ": gene ral solution examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2*x^2;" "6#*&\"\"#\"\"\"*$%\"xGF$F%" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = x^2+y^2;" "6#/*&%#dyG\"\"\"%#dx G!\"\",&*$%\"xG\"\"#F&*$%\"yGF,F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "de := 2*x^2* diff(y(x),x)=x^2+y(x)^2;\ndesolveHG(de,y(x),info=true);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/,$*(\"\"#\"\"\")%\"xGF(F)-%%diffG6$-%\"yG 6#F+F+F)F),&*$F*F)F)*$)F/F(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%IHomogeneous~DE~for~which~substit uting~~~G/%\"yG*&%#_vG\"\"\"%\"xGF(%-~~gives~.~.~G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(%\"xG!\"\"F),$*&\"\"#F(-F%6$*&F(F (,(F(F(*$)%#_vGF-F(F(*&F-F(F4F(F*F*F4F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"xG,& *&\"\"#\"\"\",&%#_vGF+F+!\"\"F.F.&%\"CG6#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"xG,& *(\"\"#\"\"\"F'F+,&%\"yG!\"\"F'F+F.F+&%\"CG6#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*( F'\"\"\",(-%#lnGF&!\"\"\"\"#F)&%\"CG6#F)F)F),&F+F-F/F)F-" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolv e(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*(F'\"\" \",(\"\"#!\"\"-%#lnGF&F)%$_C1GF)F),&F-F)F/F)F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx = y^2;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*$%\"yG\"\"#" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "de := x^2* diff(y(x),x) = y(x)^2;\ndesolveHG(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&)%\"xG\"\"#\"\"\"-%%diffG6$-%\"yG6#F(F(F**$)F. F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%IHomogeneous~DE~for~which~substituting~~~G/%\"yG*&%#_v G\"\"\"%\"xGF(%-~~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$I ntG6$*&\"\"\"F(%\"xG!\"\"F)-F%6$*&F(F(*&%#_vGF(,&F/F(F(F*F(F*F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%#lnG6#%\"xG,(-F%6#%#_vG!\"\"-F%6#,&F+\"\"\"F0F,F0&%\"CG6#F0F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%#lnG6#%\"xG,(-F%6#*&%\"yG\"\"\"F'!\"\"F.-F%6#,$*&,&F,F.F'F-F-F'F.F .F-&%\"CG6#F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG**&%\"CG6#\"\"#\"\"\"%\"yG!\"\"F$F*,&*&F+F*F $F,F*F*F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*(&%\"CG6#\"\"#\"\"\"F'F.,&F'F.F*!\" \"F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "This differential equation can also be solved by separating the va riables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "desolveSV(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DThe~DE~has~sep arable~variables~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*& \"\"\"F(*$)%\"yG\"\"#F(!\"\"F+-F%6$*&F(F(*$)%\"xGF,F(F-F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\" \"\"F&%\"yG!\"\"F(,&*&F&F&%\"xGF(F(&%\"CG6#F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$ *&F'\"\"\",&F*!\"\"*&&%\"CG6#F*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#/-%\"yG6#%\"xG*&F'\"\"\",&F)F)*&%$_C1G F)F'F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl e 3" }}{PARA 257 "" 0 "" {TEXT -1 3 " ( " }{XPPEDIT 18 0 "x^2-y^2;" "6 #,&*$%\"xG\"\"#\"\"\"*$%\"yGF&!\"\"" }{TEXT -1 2 ") " }{XPPEDIT 18 0 " dy/dx = 2*x*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(\"\"#F&%\"xGF&%\"yGF&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "de := (x^2-y(x)^2)*diff(y(x),x) = 2*x*y(x);\ndes olveHG(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&,&*$ )%\"xG\"\"#\"\"\"F,*$)-%\"yG6#F*F+F,!\"\"F,-%%diffG6$F/F*F,,$*(F+F,F*F ,F/F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%IHomogeneous~DE~for~which~substituting~~~G/%\"yG*&%#_v G\"\"\"%\"xGF(%-~~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$I ntG6$*&\"\"\"F(%\"xG!\"\"F),$-F%6$*(,&*$)%#_vG\"\"#F(F(F(F*F(F2F*,&F(F (F0F(F*F2F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%#lnG6#%\"xG,(-F%6#,&\"\"\"F,*$)%#_vG\"\"#F,F,!\"\" -F%6#F/F,&%\"CG6#F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"xG,(-F%6#*&,&*$)F'\"\"#\"\"\"F0* $)%\"yGF/F0F0F0F'!\"#!\"\"-F%6#*&F3F0F'F5F0&%\"CG6#F0F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG**& %\"CG6#\"\"#\"\"\",&F*F**&F$!\"#%\"yGF)F*!\"\"F.F*F$F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#% \"xG,&*&#\"\"\"\"\"#F+&%\"CG6#F,F+F+*&F*F+*$,&*$)F-F,F+F+*&\"\"%F+)F'F ,F+!\"\"F*F+F+/F$,&*&F*F+F-F+F+*&#F+F,F+F1F+F8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dsolve(de); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG,$*(\"\"#!\"\"%$_C1G F+,&\"\"\"F.*$,&F.F.*(\"\"%F.)F,F*F.)F'F*F.F+#F.F*F.F.F./F$,$*(F*F+F,F +,&F.F.F/F+F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam ple 4" }}{PARA 257 "" 0 "" {TEXT -1 2 "( " }{XPPEDIT 18 0 "3*x-2*y;" " 6#,&*&\"\"$\"\"\"%\"xGF&F&*&\"\"#F&%\"yGF&!\"\"" }{TEXT -1 3 " ) " } {XPPEDIT 18 0 "dy/dx = 4*x-3*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&\"\"%F &%\"xGF&F&*&\"\"$F&%\"yGF&F(" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "de := (3*x-2*y(x))*diff(y(x),x) = 4*x-3*y(x);\ndes olveHG(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&,&*& \"\"$\"\"\"%\"xGF*F**&\"\"#F*-%\"yG6#F+F*!\"\"F*-%%diffG6$F.F+F*,&*&\" \"%F*F+F*F**&F)F*F.F*F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%IHomogeneous~DE~for~which~substitutin g~~~G/%\"yG*&%#_vG\"\"\"%\"xGF(%-~~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(%\"xG!\"\"F),$*&#F(\"\"#F(-F%6$*&,& \"\"$F**&F.F(%#_vGF(F(F(,(F.F(*&F3F(F5F(F**$)F5F.F(F(F*F5F(F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%#lnG6#%\"xG,&*&#\"\"\"\"\"#F+-F%6#*&,&%#_vGF+F+!\"\"F+,&F1F+F,F2F+ F+F2&%\"CG6#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"xG,&*&#\"\"\"\"\"#F+-F%6#*(,&*&F,F+F 'F+F+%\"yG!\"\"F+,&F2F3F'F+F+F'!\"#F+F3&%\"CG6#F+F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG*&&%\"CG 6#\"\"#\"\"\",(F)F**(\"\"$F*%\"yGF*F$!\"\"F/*&F$!\"#F.F)F*#F/F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $/-%\"yG6#%\"xG,&*(\"\"$\"\"\"\"\"#!\"\"F'F+F+*&#F+F,F+*$,&*$)F'F,F+F+ *&\"\"%F+)&%\"CG6#F,F,F+F+F/F+F+/F$,&*(F*F+F,F-F'F+F+*&#F+F,F+F0F+F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG* &,&**\"\"$\"\"\"\"\"#!\"\"F'F,%$_C1GF,F,*&F-F.,&*&)F/F-F,)F'F-F,F,\"\" %F,#F,F-F.F,F/F./F$*&,&**F+F,F-F.F'F,F/F,F,*&F-F.F1F6F,F,F/F." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 257 " " 0 "" {TEXT -1 4 " ( " }{XPPEDIT 18 0 "y+x;" "6#,&%\"yG\"\"\"%\"xGF% " }{TEXT -1 3 " ) " }{XPPEDIT 18 0 "dy/dx = y-x;" "6#/*&%#dyG\"\"\"%#d xG!\"\",&%\"yGF&%\"xGF(" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "de := (y(x)+x)*diff(y(x),x) = y(x)-x;\ndesolveHG(de,i nfo=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&,&-%\"yG6#%\"xG \"\"\"F+F,F,-%%diffG6$F(F+F,,&F(F,F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%IHomogeneous~DE ~for~which~substituting~~~G/%\"yG*&%#_vG\"\"\"%\"xGF(%-~~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(%\"xG!\"\"F),$-F %6$*&,&%#_vGF(F(F(F(,&F(F(*$)F0\"\"#F(F(F*F0F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"xG,( *&#\"\"\"\"\"#F+-F%6#,&F+F+*$)%#_vGF,F+F+F+!\"\"-%'arctanG6#F2F3&%\"CG 6#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"xG,(*&#\"\"\"\"\"#F+-F%6#*&,&*$)F'F,F+F+*$) %\"yGF,F+F+F+F'!\"#F+!\"\"-%'arctanG6#*&F5F+F'F7F7&%\"CG6#F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%#lnG6#%\"xG,(*&#\"\"\"\"\"#F+-F%6#*&,&*$)F'F,F+F+*$)-%\"yGF&F,F+F+ F+F'!\"#F+!\"\"-%'arctanG6#*&F5F+F'F8F8&%\"CG6#F+F+" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dsolve(de) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&#\"\"\"\"\"#F'-%#lnG6#*&,&*$ )%\"xGF(F'F'*$)-%\"yG6#F0F(F'F'F'F0!\"#F'!\"\"-%'arctanG6#*&F3F'F0F7F7 -F*F5F7%$_C1GF7\"\"!" }}}{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 257 "" 0 "" {TEXT -1 2 "( " }{XPPEDIT 18 0 "1+x^2; " "6#,&\"\"\"F$*$%\"xG\"\"#F$" }{TEXT -1 2 ") " }{XPPEDIT 18 0 "dy/dx \+ = x*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"xGF&%\"yGF&" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "de := x^2*diff(y(x),x) = x^2+3*x*y(x)+y(x)^2;\ndesolveHG(de,info =true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&)%\"xG\"\"#\"\"\"- %%diffG6$-%\"yG6#F(F(F*,(*$F'F*F**(\"\"$F*F(F*F.F*F**$)F.F)F*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %%IHomogeneous~DE~for~which~substituting~~~G/%\"yG*&%#_vG\"\"\"%\"xGF( %-~~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F (%\"xG!\"\"F)-F%6$*&F(F(,(F(F(*&\"\"#F(%#_vGF(F(*$)F1F0F(F(F*F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%#lnG6#%\"xG,&*&\"\"\"F*,&%#_vGF*F*F*!\"\"F-&%\"CG6#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#l nG6#%\"xG,&*&,&%\"yG\"\"\"F'F,!\"\"F'F,F-&%\"CG6#F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#% \"xG,$*(F'\"\"\",(-%#lnGF&F*F*F*&%\"CG6#F*!\"\"F*,&F,F*F.F1F1F1" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*(F '\"\"\",(F*F*-%#lnGF&F*%$_C1GF*F*,&F,F*F.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 "desolveHG" }{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 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "2*x^2;" "6#*&\"\"#\"\"\"*$%\"xGF$F%" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = x^2+y^2;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*$%\"xG \"\"#F&*$%\"yGF,F&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " y(1) = 1/2" "6# /-%\"yG6#\"\"\"*&F'F'\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "de := 2*x^2 *diff(y(x),x)=x^2+y(x)^2;\nic := y(1)=1/2;\ndesolveHG(\{de,ic\},info=t rue);\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d eG/,$*(\"\"#\"\"\")%\"xGF(F)-%%diffG6$-%\"yG6#F+F+F)F),&*$F*F)F)*$)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%%IHomogeneous~DE~for~which~substituting~~~G/%\"yG*&%#_vG\"\"\"% \"xGF(%-~~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*& \"\"\"F(%\"xG!\"\"F),$*&\"\"#F(-F%6$*&F(F(,(F(F(*$)%#_vGF-F(F(*&F-F(F4 F(F*F*F4F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%#lnG6#%\"xG,&*&\"\"#\"\"\",&%#_vGF+F+!\"\"F.F.&%\" CG6#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"xG,&*(\"\"#\"\"\"F'F+,&%\"yG!\"\"F'F+F.F+&% \"CG6#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~c ondition~.~.~~G/&%\"CG6#\"\"\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*(F'\"\"\",&-%#ln GF&F)\"\"#F)F),&F+F)\"\"%F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*(9$\"\"\",&-%#lnG6#F-F.\"\"#F. 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sDqo!erF$Fjh_l7$$\"39+++6w*3e\"F1$\"3l " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 7" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = sin(y/x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%$sinG6#*&%\"yGF& %\"xGF(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 5" "6#/-%\"yG6#\"\" \"\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "de := diff(y(x),x)=sin(y(x)^2/x^2); \nic := y(1)=5;\nsoln := desolveHG(\{de,ic\},y(x));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,-%$sinG6#*&F)\"\"#F,! \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG/-%#lnG6#%\"xG-%$intG6$*&\"\" \"F.,&-%$sinG6#*$)%#_vG\"\"#F.F.F5!\"\"F7/F5;\"\"&*&-%\"yGF(F.F)F7" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 275 "plot1:=DEtools[DEplot](de,y(x),x=1..4,y=4.6..5.8,arrows=medium, \n dirgrid=[30,30],color=COLOR(RGB,.5,0,1)):\nplot2:= 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45.000000 0 0 "Curve 1" "Curve 2" }}}}{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 -1 13 "More examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = x*y/(x^2+y^2);" "6#/*&%# dyG\"\"\"%#dxG!\"\"*(%\"xGF&%\"yGF&,&*$F*\"\"#F&*$F+F.F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 2" "6#/-%\"yG6#\"\"\"\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "de := diff(y(x),x)=x*y(x)/(x^2+y(x)^2);\nic := y(1)=2 ;\ndesolveHG(\{de,ic\},info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#deG/-%%diffG6$-%\"yG6#%\"xGF,*(F,\"\"\"F)F.,&*$)F,\"\"#F.F.*$)F)F2F .F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%IHomogeneous~DE~for~which~substituting~~~G/%\"yG*&%#_vG\"\"\"% \"xGF(%-~~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*& \"\"\"F(%\"xG!\"\"F),$-F%6$*&,&F(F(*$)%#_vG\"\"#F(F(F(F2!\"$F2F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%#lnG6#%\"xG,&*&#\"\"\"\"\"#F+*&,&*(F,F+-F%6#%#_vGF+)F2F,F+F+F+!\" \"F+F2!\"#F+F4&%\"CG6#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"xG,&*&#\"\"\"\"\"#F+*&,&*( F,F+-F%6#*&%\"yGF+F'!\"\"F+)F3F,F+F4*$)F'F,F+F+F+F3!\"#F+F+&%\"CG6#F+F +" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition~ .~.~~G/&%\"CG6#\"\"\",&#F(\"\")!\"\"-%#lnG6#\"\"#F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG *&-%)LambertWG6#,$*&#\"\"\"\"\"%F/*&-%$expG6#F.F/)F'\"\"#F/F/F/#!\"\"F 6F'F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&-%)LambertWG6#,$*&#\"\"\"\"\"%F/*&-%$expG6#F.F/) F'\"\"#F/F/F/#!\"\"F6F'F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = 2*x*y/(x^2+y^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"**\"\"#F&% \"xGF&%\"yGF&,&*$F+F*F&*$F,F*F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y (1) = 2" "6#/-%\"yG6#\"\"\"\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "de := diff(y (x),x)=2*x*y(x)/(x^2+y(x)^2);\nic := y(1)=2;\ndesolveHG(\{de,ic\},y(x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF, ,$**\"\"#\"\"\"F,F0F)F0,&*$)F,F/F0F0*$)F)F/F0F0!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&#\"\"$\"\"%\"\"\"*&F+!\"\",&\"\"*F,*&\" #;F,)F'\"\"#F,F,#F,F4F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "dsolve(\{de,ic\},y(x));\nsimplify(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&#\"\"$\"\"%\"\" \"*(F*F,F+!\"\",&F,F,*(\"#;F,\"\"*F.F'\"\"#F,#F,F3F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&#\"\"$\"\"%\"\"\"*&F+!\"\",&\"\"*F,* &\"#;F,)F'\"\"#F,F,#F,F4F," }}}{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 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2*x^2;" "6#*&\"\"#\"\"\"*$%\"xGF$F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = a*x*y/(x^2+y^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"**%\"aGF&% \"xGF&%\"yGF&,&*$F+\"\"#F&*$F,F/F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 2" "6#/-%\"yG6#\"\"\"\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "de := diff (y(x),x)=a*x*y(x)/(x^2+y(x)^2);\nic := y(1)=2;\ndesolveHG(\{de,ic\},y( x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG F,**%\"aG\"\"\"F,F/F)F/,&*$)F,\"\"#F/F/*$)F)F3F/F/!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG**)\"\"#,$*&\"\"\"F*,&%\"aGF*F*!\"\"F-F-F*),&F,F- \"\"&F*,$*(F'F-F,F*F+F-F*F*)*&-%\"yG6#F$F*F$F-F)F*),$*&,(*&F,F*)F$F'F* F**$)F5F'F*F-*$F=F*F-F*F$!\"#F-,$*(F'F-F,F*F+F-F-F*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 59 "In questi ons 1 to 9, solve the given initial value problem." }}{PARA 0 "" 0 "" {TEXT -1 116 "In each case plot the graph of the solution along with t he associated direction field for the differential equation." }}{PARA 0 "" 0 "" {TEXT -1 24 "You may need to use the " }{TEXT 0 12 "implicit plot" }{TEXT -1 19 " function from the " }{TEXT 0 5 "plots" }{TEXT -1 9 " package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 4 "Note" }{TEXT -1 67 ": When setting up the differential equation to be solved by either " }{TEXT 0 6 "dsolve" }{TEXT -1 4 " or " } {TEXT 0 9 "desolveHG" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 4 " Use " }{TEXT 0 12 "diff(y(x),x)" }{TEXT -1 37 " to replace the mathema tical symbol " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 18 "Replace y by y(x) " } {TEXT 266 49 "everywhere y appears in the differential equation" } {TEXT -1 1 "." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 " " 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx = x+y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&%\"xGF&% \"yGF&" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "y(1) = 1" "6#/-%\"yG6#\"\" \"F'" }{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 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "x*y^2;" "6#*&%\"xG\"\"\"*$%\"yG\"\"#F%" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = y^3-x^3;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*$%\"yG \"\"$F&*$%\"xGF,F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 2" "6#/-% \"yG6#\"\"\"\"\"#" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "___ __________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "dy/dx = (y^2-3*x*y)/(y^2-x^2),y(1) = 1/10;" "6$/*&%#dy G\"\"\"%#dxG!\"\"*&,&*$%\"yG\"\"#F&*(\"\"$F&%\"xGF&F,F&F(F&,&*$F,F-F&* $F0F-F(F(/-F,6#F&*&F&F&\"#5F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "dy/dx = (x^2+y^2)/(x*y-5*x^2),y(1) \+ = 0;" "6$/*&%#dyG\"\"\"%#dxG!\"\"*&,&*$%\"xG\"\"#F&*$%\"yGF-F&F&,&*&F, F&F/F&F&*&\"\"&F&*$F,F-F&F(F(/-F/6#F&\"\"!" }{TEXT -1 4 " " }} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }} {PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "``(x^4+y^4)" "6#-%!G 6#,&*$%\"xG\"\"%\"\"\"*$%\"yGF)F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/ dx+2*x^3*y = 0,y(1) = 1;" "6$/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"#F'*$% \"xG\"\"$F'%\"yGF'F'\"\"!/-F/6#F'F'" }{TEXT -1 4 " " }}{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 3 "Q6 " }} {PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "``(x^2+x*y+y^2);" "6 #-%!G6#,(*$%\"xG\"\"#\"\"\"*&F(F*%\"yGF*F**$F,F)F*" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+y^2 = 0,y(1) = 1;" "6$/,&*&%#dyG\"\"\"%#dxG!\"\"F '*$%\"yG\"\"#F'\"\"!/-F+6#F'F'" }{TEXT -1 4 " " }}{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 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "dy/dx = y/x+exp(y/x),y(1) = 1;" "6$ /*&%#dyG\"\"\"%#dxG!\"\",&*&%\"yGF&%\"xGF(F&-%$expG6#*&F+F&F,F(F&/-F+6 #F&F&" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________ ________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q8 " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "dy/dx = ``(y/x)*ln(y/x),y(1) = 1;" "6$/*&%#dyG\"\"\"%#dxG!\"\"*& -%!G6#*&%\"yGF&%\"xGF(F&-%#lnG6#*&F.F&F/F(F&/-F.6#F&F&" }{TEXT -1 4 " \+ " }}{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 3 "Q9 " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "dy/dx = cos h(y/x)+y/x,y(1) = 0;" "6$/*&%#dyG\"\"\"%#dxG!\"\",&-%%coshG6#*&%\"yGF& %\"xGF(F&*&F.F&F/F(F&/-F.6#F&\"\"!" }{TEXT -1 4 " " }}{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 73 "(a) Construct a first order homogeneous d ifferential equation, for which " }{TEXT 0 6 "dsolve" }{TEXT -1 4 " or " }{TEXT 0 9 "desolveHG" }{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 chosen 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 solution along with the direction field a ssociated with the differential equation." }}{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 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }