{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 "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Tim es" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 268 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 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 "Tim es" 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 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 52 "Solution of 2nd order DEs with a \+ \"missing variable\" " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, \+ Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 10.10.2 007" }}{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 262 7 "DEsol.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 similar to the one that follows, where the file path gives \+ its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\ \\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 75 "A procedure for solving 2nd order DE's by substituting for the derivative: " }{TEXT 0 9 "desolveMV" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "desolveMV: \+ usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 263 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 17 " \+ desolveMV( de )" }}{PARA 0 "" 0 "" {TEXT -1 52 " desolveMV( de ,y(x) )\n desolveMV( \{de,cnstrts \}) " }}{PARA 0 "" 0 "" {TEXT -1 34 " d esolveMV( \{de,cnstrts \},y(x) ) " }{TEXT 265 1 "\n" }{TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 9 " de - " }{TEXT -1 70 " \+ a 2nd order linear differential equation with constant coefficients," }}{PARA 0 "" 0 "" {TEXT -1 65 " that is, one which \+ can be written in the form" }}{PARA 257 "" 0 "" {TEXT -1 4 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = F(y, dy/dx);" "6#/*(%\"dG\"\"#%\"yG\"\" \"*&F%F(*$%\"xGF&F(!\"\"-%\"FG6$F'*&%#dyGF(%#dxGF," }{TEXT -1 6 " or \+ " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = F(x,dy/dx)" "6#/*(%\"dG\"\"#%\"yG\" \"\"*&F%F(*$%\"xGF&F(!\"\"-%\"FG6$F+*&%#dyGF(%#dxGF," }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 117 " with the first an d 2nd order derivatives entered as diff(y(x),x) and diff(y(x),x$2) re spectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 82 ": The dependent variable,say y, must be ent ered as y(x) everywhere in the equation" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 23 12 "cnstrts - " } {TEXT 266 86 "two constraints involving the dependent variable, or its derivative, given in the form" }}{PARA 0 "" 0 "" {TEXT 268 53 " \+ y(a)=b or D(y)(a)=b " }}{PARA 0 "" 0 "" {TEXT -1 67 " so that \{de,cnstrts\} is a set of thre e equations." }}{PARA 0 "" 0 "" {TEXT 267 2 " " }{TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 9 "des olveMV" }{TEXT -1 74 " attempts to solve a 2nd order linear differenti al equation of the forms: " }}{PARA 15 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = F(x,dy/dx)" "6#/*(%\"dG\"\"#%\"yG\"\"\" *&F%F(*$%\"xGF&F(!\"\"-%\"FG6$F+*&%#dyGF(%#dxGF," }{TEXT -1 34 ", by m eans of the substitutions: " }{XPPEDIT 18 0 "dy/dx=u" "6#/*&%#dyG\"\" \"%#dxG!\"\"%\"uG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = du /dx;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&%#duGF(%#dxGF ," }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = F(y, dy/dx)" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xG F&F(!\"\"-%\"FG6$F'*&%#dyGF(%#dxGF," }{TEXT -1 34 ", by means of the s ubstitutions: " }{XPPEDIT 18 0 "dy/dx=u" "6#/*&%#dyG\"\"\"%#dxG!\"\"% \"uG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "d^2*y/(d*x^2)=u*``(du/dy)" "6#/ *(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&%\"uGF(-%!G6#*&%#duGF(% #dyGF,F(" }{TEXT -1 3 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT 264 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 80 "info=true/false\nWith the option \"info=true\" the s teps in the solution are shown." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to a ctivate:" }{TEXT 256 1 "\n" }{TEXT -1 155 "To make the procedure activ e, open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection." }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 16 ": The procedure " }{TEXT 0 9 "desolveMV" }{TEXT -1 32 " requires the general procedure " }{TEXT 0 7 "desolve" }{TEXT -1 2 ". " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 " desolveMV: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17334 " desolveMV := proc()\n local f f,de,ic0,ic1,x0,y0,x1,y1,prntflg,initcond,startopts,\n ee,yy,xx,Opti ons,derivs,df1,df2,df3,a,b,df,yx,x,yx2,\n x2,x3,y,vars,i,lsic,t0,t1, tt,de2,de3,de4,Cs,gt,jS,j1,j2,s,p,\n soln,assignedconst,de5,de6,de7, gx,dgx,cvals,const1,const2,\n eq0,eq1,ps,ys,gotconsts,temp,drv,pxy,p x,py,hasx,hasy,goodsols,\n order,nvars;\n global C;\n\n # Remove any solutions involving \"RootOf\".\n goodsols := proc(sol::list)\n local goodlist,i;\n goodlist := NULL;\n for i from 1 t o nops(sol) do\n if indets(sol[i],'specfunc(anything,RootOf)') =\{\} then\n goodlist := goodlist,sol[i];\n end if; \n end do;\n [goodlist]; \n end proc: # of goods ols\n\n # start of main procedure desolveMV\n if nargs>0 then \n \+ ff := args[1]\n else\n error \"at least one argument must b e supplied\"\n end if;\n initcond := false;\n if type(ff,\{set(e quation),list(equation)\}) and nops(ff)=3 then\n ff := map(_u -> \+ if has(_u,D@@2) then convert(_u,diff) else _u end if,ff);\n de := op(1,ff);\n ic0 := op(2,ff);\n ic1 := op(3,ff);\n if n ot has(de,diff) then\n de := op(2,ff);\n ic0 := op(1,f f);\n end if;\n if not has(de,diff) then\n de := op( 3,ff);\n ic1 := op(2,ff);\n end if;\n initcond := tr ue;\n elif type(ff,equation) then\n de := ff;\n else\n e rror \"the 1st argument, %1, is invalid .. it should be an equation or a set (or list) of 3 equations\",ff;\n end if;\n\n startopts := 2 ;\n if nargs>1 then\n ee := args[2];\n if type(ee,function ) and nops(ee)=1 then\n yy := op(0,ee);\n xx := op(1,e e);\n if type(xx,name) and type(yy,name) then\n sta rtopts := 3;\n else\n error \"the 2nd argument, %1, has incorrect form for the dependent variable\",ee;\n end if; \n end if;\n end if;\n\n prntflg := false;\n if nargs>=star topts then\n Options:=[args[startopts..nargs]];\n if not typ e(Options,list(equation)) then\n error \"each optional argumen t must be an equation\"\n end if;\n if hasoption(Options,'in fo','prntflg','Options') then \n if prntflg<>true then prntflg := false end if;\n end if;\n if nops(Options)>0 then\n \+ error \"%1 is not a valid option for %2\",op(1,Options),procname; \n end if;\n end if;\n \n # Check out the derivatives in th e DE.\n derivs := indets(de,'specfunc(anything,diff)');\n if deriv s=\{\} then\n error \"the 1st argument, %1, is invalid .. it shou ld be a differential equation or a set (or list) containing a differen tial equation and two initial conditions\",ff;\n end if;\n nvars : = nops(indets(derivs,name));\n if nvars<>1 then\n if nvars=0 th en\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(na me)));\n if nvars<>1 then\n if nvars=0 then\n error \"t here is a problem with the dependent variable occurring in the derivat ive(s)\"\n else\n error \"there should only be one depend ent variable in the differential equation\"\n end if;\n end if; \n\n order := nops(derivs);\n if order=1 then\n error \"the \+ differential equation should have order 2\" \n elif order>2 then\n \+ error \"there are too many derivatives in the differential equatio n .. note that the differential equation should have order 2\"\n end if;\n\n (df2,df1) := selectremove(_U->has([op(_U)],diff),derivs);\n if nops(df2)<>1 or nops(df1)<>1 then \n error \"the derivative s, %1, do not make sense\",derivs;\n end if; \n (df2,df1) := (op(d f2),op(df1));\n\n # Get the arguments in the derivatives.\n if typ e(df1,function) and op(0,df1)=diff and nops(df1)=2 then\n yx := o p(1,df1);\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,df1;\n end if; \n x : = op(2,df1);\n if not type(x,name) then\n error \"the 2nd argument %1, in the derivative, %2, is invalid .. it should be the in dependent variable\",x,df1;\n end if; \n else\n error \"th e derivative, %1, does not make sense\",df1;\n end if;\n\n if type (df2,function) and nops(df2)=2 and op(0,df2)='diff' then\n (df3,x 3) := selectremove(has,\{op(df2)\},diff);\n if nops(df3)<>1 or no ps(x3)<>1 then \n error \"the derivative, %1, does not make se nse\",df2;\n end if;\n (df3,x3) := (op(df3),op(x3));\n \+ if type(df3,function) and nops(df3)=2 and op(0,df3)='diff' then\n \+ yx2 := op(1,df3);\n if not type(yx2,anyfunc(name)) then\n \+ error \"the 1st argument %1, in the derivative, %2, is inva lid .. it should be the 'unknown' dependent variable\",yx2,df3;\n \+ end if; \n x2 := op(2,df2);\n if not type(x2,name) then\n error \"the 2nd argument %1, in the derivative, %2, is invalid .. it should be the independent variable\",x2,df3;\n \+ end if; \n if not x2=x3 then\n error \"the 2nd a rguments, %1 and %2 in the derivatives %3 and %4 should be the same\", x2,x3,df2,df3;\n end if;\n else\n error \"the der ivative, %1, does not make sense\",df3;\n end if\n else\n \+ error \"the derivative, %1, does not make sense\",df2;\n end if;\n\n # Arguments in the 2 derivatives must be the same.\n if x2<>x or \+ yx2<>yx then\n error \"the differential equation contains inconsi stent arguments\"\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 b oth appear in the differential equation\",yx,y;\n end if;\n if op( 1,yx)<>x then\n error \"the derivatives do not make sense\"\n e nd 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 assigned(C) and not type(eval(C),tabl e) then\n C := table();\n WARNING(\"C has been redefined as \+ a table for use as arbitrary constants\");\n end if;\n # Find t he indices j1,j2 to use in the constants\n Cs := select(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 nops(jS)+1 do\n \+ if not member(i,jS) then j1 := i; break; end if; \n end do;\n jS := sort([j1,op(jS)]);\n for i to nops(jS)+1 do\n if not m ember(i,jS) then j2 := i; break; end if; \n end do;\n\n if initc ond then\n # Get the boundary values.\n lsic := lhs(ic0);\n \+ if type(lsic,function) and op(0,lsic)=y and nops(lsic)=1 \n \+ and type(op(1,lsic),algebraic) then\n x 0 := op(1,lsic);\n if has(x0,\{x,y\}) then\n error \+ \"the boundary conditions must not involve %1 or %2\",x,y;\n e nd if;\n y0 := rhs(ic0);\n t0 := 0; # flag for y coord or derivative\n if has(y0,\{x,y\}) then\n error \" the boundary conditions must not involve %1 or %2\",x,y;\n end if;\n elif type(lsic,function) and op(0,lsic)=D(y) and nops(lsi c)=1 \n and type(op(1,lsic),algebraic) the n\n x0 := op(1,lsic);\n if has(x0,\{x,y\}) then\n \+ error \"the boundary conditions must not involve %1 or %2\",x,y ;\n end if;\n y0 := rhs(ic0);\n t0 := 1; # fla g for y coord or derivative\n if has(y0,\{x,y\}) then\n \+ error \"the boundary conditions must not involve %1 or %2\",x,y; \n end if;\n else\n error \"boundary condition is not decipherable\"\n end if;\n\n lsic := lhs(ic1);\n i f type(lsic,function) and op(0,lsic)=y and nops(lsic)=1 \n \+ and type(op(1,lsic),algebraic) then\n x1 := o p(1,lsic);\n if has(x1,\{x,y\}) then\n error \"the \+ boundary conditions must not involve %1 or %2\",x,y;\n end if; \n y1 := rhs(ic1);\n t1 := 0; # flag for y coord or de rivative\n if has(y1,\{x,y\}) then\n error \"the bo undary conditions must not involve %1 or %2\",x,y;\n end if;\n elif type(lsic,function) and op(0,lsic)=D(y) and nops(lsic)=1 \n and type(op(1,lsic),algebraic) then\n \+ x1 := op(1,lsic);\n if has(x1,\{x,y\}) then\n \+ error \"the boundary conditions must not involve %1 or %2\",x,y;\n \+ end if;\n y1 := rhs(ic1);\n t1 := 1; # flag for y coord or derivative\n if has(y1,\{x,y\}) then\n er ror \"the boundary conditions must not involve %1 or %2\",x,y;\n \+ end if;\n else\n error \"boundary condition is not dec ipherable\"\n end if;\n\n if t0=t1 and x0=x1 then\n \+ error \"impossible boundary conditions\"\n end if;\n if sign m(x1-x0)=-1 then # swap over\n tt := x1; x1 := x0; x0 := tt;\n tt := y1; y1 := y0; y0 := tt;\n tt := t1; t1 := t0; t 0 := tt;\n end if;\n end if;\n\n de := subs(diff(yx,x$2)=dff, de);\n drv := solve(de,dff);\n if nops([drv])<>1 then\n error \"unable to isolate the 2nd derivative\"\n end if;\n \n if prnt flg then\n print(``);\n print(diff(yx,x$2)=drv);\n end if; \n\n drv := subs(\{diff(yx,x)=p,yx=y\},drv);\n pxy := indets(drv,n ame);\n hasx := member(x,pxy);\n hasy := member(y,pxy);\n \n px \+ := _p(x);\n py := _p(y);\n if not hasx then\n de2 := diff(py, y)=simplify(subs(p=py,drv)/py);\n if prntflg then\n print (``);\n print(`Substituting . . `,py=diff(yx,x),` and `,py*dif f(py,y)=diff(yx,x$2),`gives . . `);\n print(py*diff(py,y)=subs (p=py,drv));\n print(``);\n print(`Attempting to solve the DE:`);\n print(de2);\n end if;\n de3 := traperr or(desolve(de2,py,info=prntflg));\n if de3=lasterror then \n \+ error \"unable to solve initial 1st order DE for the derivative of the solution\"\n end if;\n\n if nops([de3])<>1 then\n \+ if assigned(_ChooseSolutionDE1) and type(_ChooseSolutionDE1,posint) \n and _ChooseSolutionDE1<=nops([de3]) then\n WAR NING(\"'_ChooseSolutionDE1' is set to %1\",_ChooseSolutionDE1);\n \+ de3 := op(_ChooseSolutionDE1,[de3])\n else\n \+ WARNING(\"picking one of %1 solutions of initial 1st order DE .. othe r solutions can be obtained via '_ChooseSolutionDE1'\",nops([de3]));\n de3 := op(1,[de3])\n end if;\n end if;\n \n \+ de4 := subs(py=p,de3);\n de5 := subs(\{y=yx,p=diff(yx,x)\},de 4);\n if prntflg then\n print(`Now attempting to solve th e DE:`);\n print(de5);\n end if;\n soln := traperror (desolve(de5,yx,info=prntflg));\n if soln=lasterror then\n \+ error \"unable to solve final 1st order DE\"\n end if;\n\n \+ if nops([soln])<>1 then \n if assigned(_ChooseSolutionDE2) an d type(_ChooseSolutionDE2,posint)\n and _ChooseSolutionDE2<= nops([soln]) then\n WARNING(\"'_ChooseSolutionDE2' is set t o %1\",_ChooseSolutionDE2);\n soln := op(_ChooseSolutionDE2 ,[soln])\n else\n WARNING(\"picking one of %1 solut ions of final 1st order DE .. other solutions can be obtained via '_Ch ooseSolutionDE2'\",nops([soln]));\n soln := op(1,[soln])\n \+ end if;\n end if;\n elif not hasy then\n de2 := di ff(px,x)=simplify(subs(p=px,drv));\n if prntflg then\n pr int(``);\n print(`Substituting . . `,px=diff(yx,x),` and `,dif f(px,x)=diff(yx,x$2),`gives . . `);\n print(de2);\n pr int(``);\n print(`Attempting to solve the DE:`);\n pri nt(de2);\n end if;\n de3 := traperror(desolve(de2,px,info=pr ntflg));\n if de3=lasterror then \n error \"unable to sol ve initial 1st order DE for the derivative of the solution\"\n en d if;\n\n if nops([de3])<>1 then\n if assigned(_ChooseSol utionDE1) and type(_ChooseSolutionDE1,posint)\n and _ChooseS olutionDE1<=nops([de3]) then\n WARNING(\"'_ChooseSolutionDE 1' is set to %1\",_ChooseSolutionDE1);\n de3 := op(_ChooseS olutionDE1,[de3])\n else\n WARNING(\"picking one of %1 solutions of initial 1st order DE .. other solutions can be obtain ed via '_ChooseSolutionDE1'\",nops([de3]));\n de3 := op(1,[ de3])\n end if;\n end if;\n\n de4 := subs(px=p,de3); \n\n de5 := subs(p=diff(yx,x),de4);\n if prntflg then\n \+ print(`Now attempting to solve the DE:`);\n print(de5);\n \+ end if;\n soln := traperror(desolve(de5,yx,info=prntflg));\n if soln=lasterror then\n error \"unable to solve final 1 st order DE\"\n end if;\n\n if nops([soln])<>1 then \n \+ if assigned(_ChooseSolutionDE2) and type(_ChooseSolutionDE2,posint) \n and _ChooseSolutionDE2<=nops([soln]) then\n WA RNING(\"'_ChooseSolutionDE2' is set to %1\",_ChooseSolutionDE2);\n \+ soln := op(_ChooseSolutionDE2,[soln])\n else\n \+ WARNING(\"picking one of %1 solutions of final 1st order DE .. oth er solutions can be obtained via '_ChooseSolutionDE2'\",nops([soln])); \n soln := op(1,[soln])\n end if;\n end if;\n \+ else\n error \"the DE does not have a missing variable\"\n en d if; \n\n if initcond then\n # handle a special case first\n \+ gotconsts := false;\n if x0=x1 and t0<>t1 then\n if \+ t0=1 then ps := y0; ys := y1;\n else ps := y1; ys := y0 end if ;\n eq0 := simplify(eval(subs(\{x=x0,y=ys,p=ps\},de4)));\n \+ eq1 := simplify(eval(subs(\{x=x0,y=ys\},soln)));\n\n # fi nd the indices of the constants used\n Cs := select(type,indet s(\{eq0,eq1\}),'specindex(posint,C)');\n jS := sort([op(map(gt ,Cs))]);\n if nops(jS)<>2 then\n error \"cannot det ermine the constants\"\n end if;\n j1 := jS[1];\n \+ j2 := jS[2];\n cvals := traperror([solve(\{eq0,eq1\},\{C[j 1],C[j2]\})]);\n\n if cvals<>lasterror then\n cvals := goodsols(cvals);\n if cvals<>[] then\n co nst1 := subs(cvals[1],C[j1]);\n const2 := subs(cvals[1], C[j2]);\n if type(\{const1,const1\},set(realcons)) then \n if prntflg then\n print(`from \+ the initial conditions . . `);\n print(eq0);print( eq1);print(``);\n end if;\n gotconst s := true;\n end if;\n end if;\n end \+ if;\n end if; \n\n if not gotconsts then \n if t0=0 and t1=0 then\n eq0 := simplify(eval(subs(\{x=x0,y=y0\},so ln)));\n eq1 := simplify(eval(subs(\{x=x1,y=y1\},soln)));\n else\n if lhs(soln)<>yx then \n erro r \"unable to find a solution with the given boundary conditions\"\n \+ end if;\n gx := rhs(soln);\n dgx := tr aperror(diff(gx,x));\n if dgx=lasterror then\n \+ error \"cannot find the derivative of the solution\"\n en d if;\n if t0=1 and t1=0 then\n yy := simplif y(eval(subs(x=x0,dgx)));\n eq0 := y0=yy;\n \+ yy := simplify(eval(subs(x=x1,gx)));\n eq1 := y1=yy;\n \+ elif t0=0 and t1=1 then\n yy := simplify(eval (subs(x=x0,gx)));\n eq0 := y0=yy;\n yy := \+ simplify(eval(subs(x=x1,dgx)));\n eq1 := y1=yy;\n \+ else # t0=1 and t1=1\n yy := simplify(eval(subs(x=x 0,dgx)));\n eq0 := y0=yy;\n yy := simplify (eval(subs(x=x1,dgx)));\n eq1 := y1=yy;\n end if;\n end if;\n\n if prntflg then\n print( `from the initial conditions . . `);\n print(eq0);print(eq1 );print(``);\n end if;\n \n # find the indices of \+ the constants used\n Cs := select(type,indets(\{eq0,eq1\}),'sp ecindex(posint,C)');\n jS := sort([op(map(gt,Cs))]);\n \+ if nops(jS)<>2 then\n error \"cannot determine the constan ts\"\n end if;\n j1 := jS[1];\n j2 := jS[2];\n cvals := traperror([solve(\{eq0,eq1\},\{C[j1],C[j2]\})]);\n \+ if cvals=lasterror then\n error \"unable to find a s olution with the given boundary conditions\"\n end if;\n\n \+ cvals := goodsols(cvals);\n\n if cvals=[] then\n \+ error \"unable to find a solution with the given boundary condition s\"\n end if;\n const1 := subs(cvals[1],C[j1]);\n \+ const2 := subs(cvals[1],C[j2]);\n if not type(\{const1,con st1\},set(realcons)) then\n error \"unable to find a soluti on with the given boundary conditions\"\n end if;\n end i f;\n\n if prntflg then\n print(`so that . . `);\n \+ print(C[j1] = const1);\n print(C[j2] = const2);print(``);\n \+ end if;\n soln := traperror(simplify(eval(subs(\{C[j1]=const1 ,C[j2]=const2\},soln))));\n if soln=lasterror then\n erro r \"unable to find a solution with the given boundary conditions\"\n \+ end if;\n end;\n if lhs(soln)<>yx then\n temp := subs(yx= y,soln); \n temp := isolate(temp,x);\n if type(temp,`=`) and lhs(temp)=x then\n soln := subs(y=yx,combine(rhs(temp)))=x;\n end if;\n end if;\n soln; \nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Examples are given in the follo wing sections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolve MV" }{TEXT -1 38 ": examples with the dependent varable " }{TEXT 270 1 "y" }{TEXT -1 8 " missing" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 79 "We consider some examples of sec ond order differential equations of the form: " }{XPPEDIT 18 0 "d^2*y /(d*x^2) = F(x,dy/dx);" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(! \"\"-%\"FG6$F+*&%#dyGF(%#dxGF," }{TEXT -1 2 ". " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = -2*x*(dy/dx)^2;" "6#/*(%\"dG\"\"#%\"yG \"\"\"*&F%F(*$%\"xGF&F(!\"\",$*(F&F(F+F(*&%#dyGF(%#dxGF,F&F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 1;" "6#/-%$y~'G6#\"\"!\"\"\"" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 101 "de := diff(y(x),x$2)=-2*x*diff(y(x),x)^2;\nic := y(0)=0,D(y)(0)=1;\nsoln := desolve(\{de,ic\},info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\" #,$*(F0\"\"\"F,F3)-F'6$F)F,F0F3!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5The~DE~is~not~ linearG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,$*(F.\"\"\"F*F1 )-F%6$F'F*F.F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_pG6#%\"xG-%%diffG6$ -%\"yGF'F(%&~and~G/-F*6$F%F(-F*6$F,-%\"$G6$F(\"\"#%+gives~.~.~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%#_pG6#%\"xGF*,$*(\"\"#\" \"\")F'F-F.F*F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%%solnG/-%\"y G6#%\"xG-%'arctanGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(soln,de);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%'arctanG6#%\"xG-%\"$G6$F*\"\"#,$*( F.\"\"\"F*F1)-F%6$F'F*F.F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$ *(\"\"#\"\"\",&F'F'*$)%\"xGF&F'F'!\"#F+F'!\"\"F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dsolve(\{de, ic\},y(x));\nop(\{allvalues(%)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"yG6#%\"xG,$*&-%'RootOfG6#,&*$)%#_ZG\"\"#\"\"\"F2F2F2F2-%(arctanhG 6#*&F'F2F*F2F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG- %'arctanGF&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = -``(1/(x^2))*(dy/dx)^2;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF& F(!\"\",$*&-%!G6#*&F(F(*$F+F&F,F(*$*&%#dyGF(%#dxGF,F&F(F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 0;" "6#/-%\"yG6#\"\"\"\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(1) = -2;" "6#/-%$y~'G6#\"\"\",$\"\"#!\" \"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "de := diff(y(x),x$2)=-diff(y(x),x)^2/x^2 ;\nic := y(1)=0,D(y)(1)=-2;\nsoln := desolve(\{de,ic\},info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$ F,\"\"#,$*&-F'6$F)F,F0F,!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#icG6$/-%\"yG6#\"\"\"\"\"!/--%\"DG6#F(F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5The~DE~is~not~ linearG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,$*&-F%6$F'F*F.F *!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_pG6#%\"xG-%%diffG6$-%\"yGF'F(% &~and~G/-F*6$F%F(-F*6$F,-%\"$G6$F(\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%#_pG6#%\"xGF*,$*&F'\"\"#F*!\"#!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%%%solnG/-%\"yG6#%\"xG,(*&\"\"#\"\"\"F)F-F-*&\"\"%F--%# lnG6#,&F,F-F)!\"\"F-F-F,F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(soln,de);\nsimplify(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$,(*&\"\"#\"\"\"%\"xGF*F**& \"\"%F*-%#lnG6#,&F)F*F+!\"\"F*F*F)F2-%\"$G6$F+F),$*&-F%6$F'F+F)F+!\"#F 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"%\"\"\",&\"\"#!\"\"%\"xG F'!\"#F*F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#% \"xG,,*&\"\"#\"\"\"F'F+F+*&\"\"%F+-%#lnG6#,&F+!\"\"*&F*F2F'F+F+F+F+F*F 2*&F-F+-F/6#F*F+F+*&^#!\"%F+%#PiGF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 10 "Example 3 " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = 1/x;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F( *$%\"xGF&F(!\"\"*&F(F(F+F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6# *&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 0;" "6#/-%\"yG6#\"\"\"\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(1) = \+ -2;" "6#/-%$y~'G6#\"\"\",$\"\"#!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "de := d iff(y(x),x$2)=diff(y(x),x)/x;\nic := y(1)=0,D(y)(1)=2;\nsoln := desolv e(\{de,ic\},info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%d iffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&-F'6$F)F,\"\"\"F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"\"\"\"!/--%\"DG6#F(F)\" \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%KThe~DE~does~not~have~constant~coefficientsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%d iffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#*&-F%6$F'F*\"\"\"F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Sub stituting~.~.~G/-%#_pG6#%\"xG-%%diffG6$-%\"yGF'F(%&~and~G/-F*6$F%F(-F* 6$F,-%\"$G6$F(\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %%diffG6$-%#_pG6#%\"xGF**&F'\"\"\"F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%%solnG/-%\"yG6#%\"xG,&* $)F)\"\"#\"\"\"F.F.!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(soln,de);\nsimplify(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$,&*$)%\"xG\"\"#\"\"\"F,F,! \"\"-%\"$G6$F*F+*&-F%6$F'F*F,F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ \"\"#F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*$)F'\"\"#\"\"\"F,F,!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+tan(x);" "6#,&*(%\"dG\"\"#%\"y G\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(-%$tanG6#F+F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 2;" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "`y '`(0) = 1;" "6#/-%$y~'G6#\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "de := diff(y(x),x$2)+tan(x)*diff(y(x),x)=0;\nic := y (0)=2,D(y)(0)=1;\nsoln := desolve(\{de,ic\},info=true);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"# \"\"\"*&-%$tanGF,F2-F(6$F*F-F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"#/--%\"DG6#F(F)\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%KThe~DE~does~ not~have~constant~coefficientsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F* \"\"#,$*&-%$tanGF)\"\"\"-F%6$F'F*F3!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_ pG6#%\"xG-%%diffG6$-%\"yGF'F(%&~and~G/-F*6$F%F(-F*6$F,-%\"$G6$F(\"\"#% +gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%#_pG6#%\" xGF*,$*&-%$tanGF)\"\"\"F'F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%%solnG/-%\"yG6 #%\"xG,&-%$sinGF(\"\"\"\"\"#F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(soln,de);\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%diffG6$,&-%$sinG6#%\"xG\"\" \"\"\"#F--%\"$G6$F,F.F-*&-%$tanGF+F--F&6$F(F,F-F-\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&-%$sinGF&\"\"\"\"\"#F+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 \+ " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = \+ ``(dy/dx)*(dy/dx+x);" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\" \"*&-%!G6#*&%#dyGF(%#dxGF,F(,&*&F2F(F3F,F(F+F(F(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "y(1) = 1;" "6#/-%\"yG6#\"\"\"F'" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`y '`(1) = 1" "6#/-%$y~'G6#\"\"\"F'" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "de := diff(y(x),x$2)=diff(y(x),x)/(diff(y(x),x)+x);\nic := y(1) =1,D(y)(1)=1;\ndesolve(\{de,ic\},y(x),info=true);\ng := unapply(rhs(%) ,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG -%\"$G6$F,\"\"#*&-F'6$F)F,\"\"\",&F2F4F,F4!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"\"F*/--%\"DG6#F(F)F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5The~DE~ is~not~linearG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#*&-F%6$F'F* \"\"\",&F0F2F*F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_pG6#%\"xG-%%diffG6$ -%\"yGF'F(%&~and~G/-F*6$F%F(-F*6$F,-%\"$G6$F(\"\"#%+gives~.~.~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%#_pG6#%\"xGF**&F'\"\"\",& F'F,F*F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%\"gGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,$*&#\"\"\"\"\"%F/*&,(*(\"\"#F/)9$F4F/-%)LambertWG6#*&-%$expG6# F/F/F6F/F/F/*$F5F/F/*$)F7F4F/F/F/F7!\"#F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "We can construct a numeri cal solution " }{TEXT 262 2 "gn" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "de := diff (y(x),x$2)=diff(y(x),x)/(diff(y(x),x)+x);\nic := y(1)=1,D(y)(1)=1;\ngn := desolve(\{de,ic\},y(x),x=-0.135..8,type=numeric,\n \+ method=rk78,errorcontrol=accumulative);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&-F'6$F)F ,\"\"\",&F2F4F,F4!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-% \"yG6#\"\"\"F*/--%\"DG6#F(F)F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 52 "The two solutions can be compared graphic ally . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "plot([g(x),'gn'(x)],x=-0.135..8,thickness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7S 7$$!$N\"!\"$$\"+ShoaC!#57$$\"*#\\)>B%F-$\"+)**pE+$F-7$$\"+=!\\g'>F-$\" +buZJPF-7$$\"+(zX6q$F-$\"+GP_fZF-7$$\"+BRuZaF-$\"+&R()*))fF-7$$\"+u2/' =(F-$\"+)*z5*Q(F-7$$\"*Zkwz)!\"*$\"+)>\"=M))F-7$$\"+Y2kY5FJ$\"+-D=Z5FJ 7$$\"+&QA#>7FJ$\"+o7.J7FJ7$$\"+_0D\"R\"FJ$\"+MJSG9FJ7$$\"+nE?o:FJ$\"++ OkX;FJ7$$\"+'=iSs\"FJ$\"+;*)f[=FJ7$$\"+!4B&**=FJ$\"+yEw*3#FJ7$$\"+_Wqv ?FJ$\"+p\"G^M#FJ7$$\"+Ty[XAFJ$\"+b6[.EFJ7$$\"+Bum*R#FJ$\"+IBK[GFJ7$$\" +f=+$e#FJ$\"+0s\">:$FJ7$$\"+v,JQFFJ$\"+))=_>MFJ7$$\"+n1(*=HFJ$\"+)[\\E u$FJ7$$\"+9e))yIFJ$\"+Q'\\\"RSFJ7$$\"+w!QVD$FJ$\"+bWivVFJ7$$\"+?,T@MFJ $\"+IYv1ZFJ7$$\"+qAt&f$FJ$\"+[aFj]FJ7$$\"+AW\"ev$FJ$\"+GrZ+aFJ7$$\"+]Y [GRFJ$\"+ShgudFJ7$$\"+)[Sy5%FJ$\"+!\\XX<'FJ7$$\"+s-(RE%FJ$\"+K!)*>`'FJ 7$$\"+;ZfKWFJ$\"+K\\pFpFJ7$$\"+n-!og%FJ$\"+q!4pM(FJ7$$\"+7kAxZFJ$\"+]( )=nxFJ7$$\"+rC7U\\FJ$\"+o\"=L=)FJ7$$\"+I9@D^FJ$\"+7O?c')FJ7$$\"+-]s*G& FJ$\"+G@x!4*FJ7$$\"+c$y`Y&FJ$\"+G\\yk&*FJ7$$\"+-kaCcFJ$\"+)[=.+\"!\")7 $$\"+=\"f&)z&FJ$\"+_g?\\5Fcv7$$\"+A:HifFJ$\"+!=9h4\"Fcv7$$\"+i!RM8'FJ$ \"+eD3Y6Fcv7$$\"+lqw+jFJ$\"+#)\\&e>\"Fcv7$$\"+xR%fZ'FJ$\"+?+$*[7Fcv7$$ \"+k,mWmFJ$\"+f\"y4I\"Fcv7$$\"+Lo>b#3Y\"Fcv7$$\"+E\"RdK(FJ$\"+h#o-_\"Fcv7$$\"++ c\"p[(FJ$\"+9RHu:Fcv7$$\"+QLwewFJ$\"+KfyK;Fcv7$$\"+]NCByFJ$\"+dfi*o\"F cv7$$\"\")\"\"!$\"+Epj^$\"+'R()*))fF-7$FC$\"+(*z5*Q(F-7$FH$\"+*>\"=M))F-FM7$FS$\"+p7.J7F J7$FX$\"+NJSG9FJ7$Fgn$\"+)fVck\"FJ7$F\\o$\"+<*)f[=FJ7$Fao$\"+wEw*3#FJ7 $Ffo$\"+n\"G^M#FJFjo7$F`p$\"+JBK[GFJ7$Fep$\"+.s\">:$FJFipF^q7$Fdq$\"+N '\\\"RSFJ7$Fiq$\"+dWivVFJ7$F^r$\"+JYv1ZFJ7$Fcr$\"+YaFj]FJ7$Fhr$\"+FrZ+ aFJ7$F]s$\"+QhgudFJ7$Fbs$\"+\"\\XX<'FJ7$Fgs$\"+L!)*>`'FJ7$F\\t$\"+J\\p FpFJ7$Fat$\"+u!4pM(FJ7$Fft$\"+[()=nxFJ7$F[u$\"+p\"=L=)FJ7$F`u$\"+4O?c' )FJ7$Feu$\"+I@x!4*FJ7$Fju$\"+C\\yk&*FJF^vFdvFivF^wFcwFhwF]xFbx7$Fhx$\" +(f$z49Fcv7$F]y$\"+=b#3Y\"FcvFayFfyF[zF`zFez-F\\[l6&F^[lFb[lF_[lFb[l-F d[l6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!F`bl-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 " \+ . . . and numerically . . ." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "xx := -.135;\ngn(xx);\nevalf (g(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!$N\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+RhoaC!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+RhoaC!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "xx := 0.00 00001;\ngn(xx);\nevalf(g(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x xG$\"\"\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+w&Q$QG!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+v&Q$QG!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "xx := 8;\ngn (xx);\nevalf(g(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"\")" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Epj^ " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"\"\"\"#F+*&-%)L ambertWG6#*&-%$expG6#F+F+F'F+!\"\"F'F,F+F+*&#F+\"\"%F+*&F.!\"#F'F,F+F+ F7F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 6 \+ " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = ``(1/x)*``(dy/dx)+(dy/dx)^2;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"x GF&F(!\"\",&*&-%!G6#*&F(F(F+F,F(-F06#*&%#dyGF(%#dxGF,F(F(*$*&F6F(F7F,F &F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 0;" "6#/-%\"yG6#\"\"\"\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(1) = 1/8;" "6#/-%$y~'G6# \"\"\"*&F'F'\"\")!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "de := diff(y(x),x$2)=d iff(y(x),x)/x+diff(y(x),x)^2;\nic := y(1)=0,D(y)(1)=1/8;\nsol := desol ve(\{de,ic\},info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%% diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#,&*&-F'6$F)F,\"\"\"F,!\"\"F5*$)F3F0 F5F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"\"\"\"!/- -%\"DG6#F(F)#F*\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5The~DE~is~not~linearG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"y G6#%\"xG-%\"$G6$F*\"\"#*(-F%6$F'F*\"\"\",&F2F2*&F0F2F*F2F2F2F*!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6'%2Substituting~.~.~G/-%#_pG6#%\"xG-%%diffG6$-%\"yGF'F(%&~and~G/-F*6$ F%F(-F*6$F,-%\"$G6$F(\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%#_pG6#%\"xGF**(F'\"\"\",&F,F,*&F'F,F*F,F,F,F*!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%$solG/-%\"yG6# %\"xG,&-%#lnG6#,&*$)F)\"\"#\"\"\"!\"\"\"# " 0 "" {MPLTEXT 1 0 122 "e1 := simplify(exp(rhs(sol)));\nassume(x_ -sqrt(17));\nsubs(x_=x,ln(subs(x=x_,e1)));\ng := unapply(%,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G,$*&\"#;\"\"\",&*$)%\"xG\"\"#F(F (\"#%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%#lnG6#,$*&\"#;\"\"\",&*$)9$ \"\"#F2F2\"# " 0 "" {MPLTEXT 1 0 32 "subs(y(x)=g(x),de);\nsimplif y(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%#lnG6#,$*&\"#;\" \"\",&*$)%\"xG\"\"#F-F-\"# " 0 "" {MPLTEXT 1 0 19 "plot(g(x),x=-4..4);" }} {PARA 13 "" 1 "" {GLPLOT2D 441 327 327 {PLOTDATA 2 "6%-%'CURVESG6$7]o7 $$!\"%\"\"!$\"39\"yRAs)esF!#<7$$!3iLL$e%G?yRF-$\"3Ty[gz`D7EF-7$$!3ymmm \"p0k&RF-$\"3ey,^\">F-7$$!37++]\">:F-7$$!3m****\\AHe)e$F-$\"3y7F-7$$!3Mmmm1rQGX-fX(**!#=7$$!3_LLL[9cgJF-$\"3=X&>I$eH^#)Fdo7$$!3smm mhN2-IF-$\"3gz]Q15/ZpFdo7$$!3!******\\`oz$GF-$\"3_'psTG&*Q\"eFdo7$$!3! omm;)3DoEF-$\"3'p!4fr&>.#[Fdo7$$!3?+++:v2*\\#F-$\"3?/QBg2asRFdo7$$!3BL LL8>1DBF-$\"3G%fe\"p[$4A$Fdo7$$!3kmmmw))yr@F-$\"3DU`)G8aMk#Fdo7$$!3;++ +S(R#**>F-$\"31@jJ/^0u?Fdo7$$!30++++@)f#=F-$\"3'*f$zmQJpd\"Fdo7$$!3-++ +gi,f;F-$\"33-kd^o&*f6Fdo7$$!3qmmm\"G&R2:F-$\"3%[7Gs-haG)!#>7$$!3XLLLt K5F8F-$\"33$*Q=a=Tu[F\\s7$$!3eLLL$HsV<\"F-$\"3S8'RM+=#)R#F\\s7$$!3+-++ ]&)4n**Fdo$!3awUp%*G20T!#@7$$!37PLLL\\[%R)Fdo$!3L**G$>*4'*G=F\\s7$$!3G )*****\\&y!pmFdo$!3c2RK5yN6MF\\s7$$!3Y******\\O3E]Fdo$!3)\\a2vVR`c%F\\ s7$$!3NKLLL3z6LFdo$!3h$>ky!z>:aF\\s7$$!3sLLL$)[`P(******z-6j'Fdo$!3!>b$Q,A'=W$F\\s7$$\"3q\"******4# 32$)Fdo$!3V,F\\s7$$\"3r$*****\\#y'G**Fdo$!3uxL&>(G[z))F\\t7$$\" 3G******H%=H<\"F-$\"3wye9\"ptjP#F\\s7$$\"35mmm1>qM8F-$\"3eZf$QcFs+&F\\ s7$$\"3%)*******HSu]\"F-$\"39?>3wDQ'G)F\\s7$$\"3'HLL$ep'Rm\"F-$\"3'f$R dP+^r6Fdo7$$\"3')******R>4N=F-$\"3[*e:!QlO,;Fdo7$$\"3#emm;@2h*>F-$\"3y C7.[YVk?Fdo7$$\"3]*****\\c9W;#F-$\"3cU&fl^buh#Fdo7$$\"3Lmmmmd'*GBF-$\" 3:nEF-$ \"3B8V`jsQ9[Fdo7$$\"35LLL.a#o$GF-$\"3x@)\\D`Ym!eFdo7$$\"3ammm^Q40IF-$ \"3Mvl%\\T\"yppFdo7$$\"3y******z]rfJF-$\"3Cu9/R(oOC)Fdo7$$\"3gmmmc%GpL $F-$\"3)olEb%=g.5F-7$$\"3#)*****\\LzhT$F-$\"3UB*)f_-H*4\"F-7$$\"3/LLL8 -V&\\$F-$\"3\\8[nBmt27F-7$$\"3imm;z\"G*zNF-$\"3t)eWPp$HT8F-7$$\"3=+++X hUkOF-$\"3!\\\\%fq$o%*\\\"F-7$$\"3=+++![,`u$F-$\"3UYQ.KC6$o\"F-7$$\"3= +++:oF-7$$\"37++D6EjpQF-$\"3=M]*pA`l1#F-7$$\"34++ ]2%)38RF-$\"3kOGo!*))>\\AF-7$$\"31+]i0j\"[$RF-$\"31#QToe3dN#F-7$$\"3/+ +v.UacRF-$\"3qBz2E8jvCF-7$$\"3-+](=5s#yRF-$\"31xyZ$*[s7EF-7$$\"\"%F*F+ -%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fa`l-%+AXESLABELSG6$Q\"x6\"Q!Ff`l-%%V IEWG6$;F(Fh_l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$-%#lnG6#,&*&\"#;!\"\"F'\"\"# F/#\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolveMV" }{TEXT -1 40 " : examples with the independent varable " }{TEXT 271 1 "x" }{TEXT -1 9 " missing " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 79 "We consider some examples of second order diffe rential equations of the form: " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = F(y, dy/dx);" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"-%\"FG6$F'* &%#dyGF(%#dxGF," }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)=(dy/dx)^2" " 6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*$*&%#dyGF(%#dxGF,F&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(1) = 3;" "6#/-%$y~'G6#\"\"\"\" \"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y (2)=1" "6#/-%\"yG6#\"\"#\"\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "de := diff(y(x),x$2)=diff(y(x),x)^2;\nic := y(1)=3,D(y)(1)=1;\ndesolve(\{de,ic\},y(x),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*$)-F '6$F)F,F0\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\" \"\"\"\"$/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%5The~DE~is~not~linearG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG 6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#*$)-F%6$F'F*F.\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting ~.~.~G/-%#_pG6#%\"yG-%%diffG6$-F(6#%\"xGF.%&~and~G/*&F%\"\"\"-F*6$F%F( F2-F*6$F,-%\"$G6$F.\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\"\"-%%diffG6$F%F(F)*$)F%\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 34 "dsolve(\{de,ic\},y(x)); \nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$-%#l nG6#,&*&-%$expG6#!\"$\"\"\"F'F2!\"\"*&\"\"#F2F.F2F2F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&\"\"$\"\"\"-%#lnG6#,&\"\"#F*F'!\" \"F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 \+ " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+3; " "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"$F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\" F'*&\"\"#F'%\"yGF'F'\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y (0) = 1 " "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(1) \+ = 0;" "6#/-%$y~'G6#\"\"\"\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "de := diff( y(x),x$2)+3*diff(y(x),x)+2*y(x)=0;\nic := y(0)=1,D(y)(1)=0;\ndesolve( \{de,ic\},y(x),method=missingvar,info=true);\nmap(simplify,%,symbolic) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG- %\"$G6$F-\"\"#\"\"\"*&\"\"$F2-F(6$F*F-F2F2*&F1F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(6#F+ F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,&*&\"\"$\"\"\"- F%6$F'F*F2!\"\"*&F.F2F'F2F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_pG6#%\"yG-%%d iffG6$-F(6#%\"xGF.%&~and~G/*&F%\"\"\"-F*6$F%F(F2-F*6$F,-%\"$G6$F.\"\"# %+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\" \"-%%diffG6$F%F(F),&*&\"\"$F)F%F)!\"\"*&\"\"#F)F(F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 22 "desolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(-%$expG6#!\"\"\"\"\",&F*\" \"#F.F-F--F+6#,$F'F-F.F0*&F/F--F+6#,$F'!\"#F.F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2;" "6#,&*(%\"dG\"\"#%\"yG\"\" \"*&F%F(*$%\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+ y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF'\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "y(0) = 2;" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`y '`(1) = 1;" "6#/-%$y~'G6#\"\"\"F'" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "de := diff(y(x),x$2)+2*diff(y(x),x)+y(x)=0;\nic := y (0)=2,D(y)(0)=1;\ndesolve(\{de,ic\},y(x),method=missingvar,info=true); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-% \"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2F*F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"#/--%\"DG6#F(F)\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,&*&F.\"\"\"-F%6$F'F*F1!\"\"F' F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_pG6#%\"yG-%%diffG6$-F(6#%\"xGF .%&~and~G/*&F%\"\"\"-F*6$F%F(F2-F*6$F,-%\"$G6$F.\"\"#%+gives~.~.~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\"\"-%%diffG6$F%F(F) ,&*&\"\"#F)F%F)!\"\"F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&-%$expG6#,$F'!\"\"\"\"#*(\"\"$\"\"\"F)F1F'F1F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }} {PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+y = 0; " "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)F(F)\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 1;" "6#/-%$y~'G6#\"\"!\"\" \"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "de := diff(y(x),x$2)+y(x)=0;\nic := y(0)=0 ,D(y)(0)=1;\ndesolve(\{de,ic\},y(x),method=missingvar,info=true);\nsim plify(%,assume=real);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%d iffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"F*F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%d iffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,$F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~. ~.~G/-%#_pG6#%\"yG-%%diffG6$-F(6#%\"xGF.%&~and~G/*&F%\"\"\"-F*6$F%F(F2 -F*6$F,-%\"$G6$F.\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*&-%#_pG6#%\"yG\"\"\"-%%diffG6$F%F(F),$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 21 "dsolve( \{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$s inGF&" }}}{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 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = 1/ ``(dy/dx);" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&F(F(-%! G6#*&%#dyGF(%#dxGF,F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6 #/-%\"yG6#\"\"!F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 3" "6# /-%$y~'G6#\"\"!\"\"$" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "de := diff(y(x),x$2)=1/diff(y(x),x);\nic := y(0)=0,D (y)(0)=3;\ndesolve(\{de,ic\},y(x),method=missingvar,info=true);\ng := \+ unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6 $-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&\"\"\"F2-F'6$F)F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#*&\"\"\"F0-F%6$F'F*!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 '%2Substituting~.~.~G/-%#_pG6#%\"yG-%%diffG6$-F(6#%\"xGF.%&~and~G/*&F% \"\"\"-F*6$F%F(F2-F*6$F,-%\"$G6$F.\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\"\"-%%diffG6$F%F(F)*&F)F)F%!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,& \"\"*!\"\"*&#\"\"\"\"\"$F1*$*$),&*&\"\"#F19$F1F1F-F1F2F1#F1F8F1F1F(F(F (" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "We \+ can also construct a numerical solution . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "de := diff( y(x),x$2)=1/diff(y(x),x);\nic := y(0)=0,D(y)(0)=3;\ngn := desolve(\{de ,ic\},y(x),x=-4.5..6,type=numeric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&\"\"\"F2-F'6$F)F,!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6# F(F)\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 " . . . and compare the two solutions graphically. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot([g(x),'gn'(x)],x=-4.5..6,thick ness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6& -%'CURVESG6%7S7$$!3++++++++X!#<$!\"*\"\"!7$$!3E++D\"))H6F%F*$!3'RoAWlp n*))F*7$$!3O+vo97*>2%F*$!3c#HGpq+gt)F*7$$!3)***\\(o&*Q![QF*$!3c3(yOO'o .&)F*7$$!31+]7)>-Ei$F*$!3^(zZ%o$[^A)F*7$$!3'**\\P4!pB)R$F*$!3ZsxVUZn4z F*7$$!39+vo\\:A!>$F*$!3-U,4+^u'e(F*7$$!3)**\\(=-P$[(HF*$!3%*HryjvG\"*)G+Kt$Ffp7$$\"3kP+v$4_E.*!#>$\"3Wglc4'[Ls#Ffp7 $$\"3W0+DJXF`JFfp$\"3ONf!\\McOi*Ffp7$$\"3E-]ila[>_Ffp$\"3)Qp-j/4/h\"F* 7$$\"3#o**\\P4t\"[uFfp$\"3y&3(\\Il]CBF*7$$\"3s%*\\7yJ:j(*Ffp$\"37#*o') *)*)[#3$F*7$$\"31+vo9]$y<\"F*$\"3K5]%*))=^bPF*7$$\"3++]73=[&R\"F*$\"35 A0M#=Ef\\%F*7$$\"3(*****\\UAL?;F*$\"3yUyPGELv_F*7$$\"3C**\\iDXIS=F*$\" 3%Ra!p-l\"=0'F*7$$\"3'**\\7.-RJ0#F*$\"3zW\\)>/Ug\"oF*7$$\"3u**\\PRaX*G #F*$\"3S0tFVfBzwF*7$$\"3E****\\_iz,DF*$\"3qIS_(>*yn%)F*7$$\"3T++v$R:&G FF*$\"3kKb3-Z.B$*F*7$$\"3_*\\7GjcR$HF*$\"3LM,az&p4,\"!#;7$$\"3;++Dr\"e &eJF*$\"3I#G\"4>WA)4\"Fdv7$$\"3%))\\7`r!*)pLF*$\"3c$H*)=!=]\"=\"Fdv7$$ \"3!3]il6%z!f$F*$\"3gVZ^IQvp7Fdv7$$\"3*))**\\(ovw1QF*$\"3?#3n#y#4sN\"F dv7$$\"32*\\7yvrG.%F*$\"3-+\"H$*)H**\\9Fdv7$$\"3++++lpj]UF*$\"3&Hc]r([ _S:Fdv7$$\"3m**\\(=MLLZ%F*$\"3x&3f?4zUj\"Fdv7$$\"3g+DJ!o&=%p%F*$\"3O%Q !zFKTGFdv7$$\"3W+++b@vP`F*$\"3G/A&Rkf\"4?Fdv7$$\"3!)*\\7`Jf&fbF* $\"3wX#>kK*43@Fdv7$$\"3!4](opq&=x&F*$\"3s$>>&QZ#Q?#Fdv7$$\"\"'F-$\"3h( 3pk)H!yI#Fdv-%'COLOURG6&%$RGBG$\"#5!\"\"$F-F-Fb[l-%*THICKNESSG6#\"\"\" -F$6%7S7$$!#XFa[l$!+,+++!*F,7$$!+\"))H6F%F,$!+a'pn*))F,7$$!+:7*>2%F,$! +22+O()F,7$$!+d*Q![QF,$!+kjo.&)F,7$$!+)>-Ei$F,$!+o$[^A)F,7$$!+,pB)R$F, $!+UZn4zF,7$$!+]:A!>$F,$!+,^u'e(F,7$$!+-P$[(HF,$!+jv_F,$\"+[!4/h\"F,7$$\"*4t\"[uF,$\"+Hl]CBF,7$$\"*=`Jw*F,$\"+!**) [#3$F,7$$\"+:]$y<\"F,$\"+!*=^bPF,7$$\"+3=[&R\"F,$\"+#=Ef\\%F,7$$\"+UAL ?;F,$\"+FELv_F,7$$\"+EXIS=F,$\"+/l\"=0'F,7$$\"+?!RJ0#F,$\"+T?/;oF,7$$ \"+RaX*G#F,$\"+UfBzwF,7$$\"+_iz,DF,$\"+'>*yn%)F,7$$\"+%R:&GFF,$\"+.Z.B $*F,7$$\"+Lm&R$HF,$\"+!ep4,\"!\")7$$\"+r\"e&eJF,$\"+>WA)4\"Fjfl7$$\"+: 2*)pLF,$\"+-=]\"=\"Fjfl7$$\"+Fjfl7$$\"+b@vP`F,$\"+W'f\"4?Fjfl7$$ \"*Kf&fbFjfl$\"+H$*43@Fjfl7$$\"*2d=x&Fjfl$\"+RZ#Q?#Fjfl7$Fgz$\"+')H!yI #Fjfl-F\\[l6&F^[lFb[lF_[lFb[l-Fd[l6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fh[m -%%VIEWG6$;F[\\lFgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y (x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"$!\"\", &*&\"\"#\"\"\"F'F/F/\"\"*F/#F*F.F/F0F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 10 "Example 6 " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = 2*y*``(dy/dx);" "6#/*(%\"dG\"\"#%\"yG\" \"\"*&F%F(*$%\"xGF&F(!\"\"*(F&F(F'F(-%!G6#*&%#dyGF(%#dxGF,F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 1" "6#/-%$y~'G6#\"\"!\"\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "de := diff(y(x),x$2)=2*y(x)*diff(y(x),x);\nic : = y(0)=0,D(y)(0)=1;\ndesolve(\{de,ic\},y(x),method=missingvar,info=tru e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG- %\"$G6$F,\"\"#,$*(F0\"\"\"F)F3-F'6$F)F,F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%d iffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,$*(F.\"\"\"F'F1-F%6$F'F*F1F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 '%2Substituting~.~.~G/-%#_pG6#%\"yG-%%diffG6$-F(6#%\"xGF.%&~and~G/*&F% \"\"\"-F*6$F%F(F2-F*6$F,-%\"$G6$F.\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\"\"-%%diffG6$F%F(F),$*(\"\"#F)F(F )F%F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$tanGF&" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 7 " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = 2*y*``(dy/dx)^2;" "6#/ *(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*(F&F(F'F(-%!G6#*&%#dyGF( %#dxGF,F&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\" \"!F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 1" "6#/-%$y~'G6#\" \"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "de := diff(y(x),x$2)=2*y(x)*diff(y (x),x)^2;\nic := y(0)=0,D(y)(0)=1;\ndesolve(\{de,ic\},y(x),method=miss ingvar,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6 $-%\"yG6#%\"xG-%\"$G6$F,\"\"#,$*(F0\"\"\"F)F3)-F'6$F)F,F0F3F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F) \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,$*(F.\"\"\"F'F1 )-F%6$F'F*F.F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_pG6#%\"yG-%%diffG6$-F(6 #%\"xGF.%&~and~G/*&F%\"\"\"-F*6$F%F(F2-F*6$F,-%\"$G6$F.\"\"#%+gives~.~ .~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\"\"-%%diffG6 $F%F(F),$*(\"\"#F)F(F))F%F/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 79 "fn := proc(x)\n local \+ y;\n fsolve(sqrt(Pi)/2*erf(y)=x,y=x/(1-x^4));\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can also con struct a numerical solution " }{TEXT 262 2 "gn" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "de := diff(y(x),x$2)=2*y(x)*diff(y(x),x)^2;\nic := y(0)=0,D(y)(0) =1;\ngn := desolveK2(\{de,ic\},y(x),x=-sqrt(Pi)/2..sqrt(Pi)/2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$ F,\"\"#,$*(F0\"\"\"F)F3)-F'6$F)F,F0F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The two solutions can b e compared graphically . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "plot(['gn'(x),'fn'(x)],x=-s qrt(Pi)/2..sqrt(Pi)/2,color=[red,green],\n \+ thickness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7ao7$$!+T#pA'))!#5$!+$pRR+%!\"*7$$!+uf>]))F *$!+/8_kAF-7$$!+4F7Q))F*$!+AiM>@F-7$$!+V%\\g#))F*$!+?_OI?F-7$$!+yh(R\" ))F*$!+C9>l>F-7$$!+Z'H)*y)F*$!+Mc:q=F-7$$!+;Jol()F*$!+4sC+=F-7$$!+a+R< ()F*$!+jC!yp\"F-7$$!+#*p4p')F*$!+=5-A;F-7$$!+o3^s&)F*$!+Y&Q-^\"F-7$$!+ VZ#fZ)F*$!+(y?pU\"F-7$$!+.m%yI)F*$!+p&\\oJ\"F-7$$!+i%o(R\")F*$!+g6rJ7F -7$$!+lbshxF*$!+!fTr3\"F-7$$!+bn<\"Q(F*$!+zNso(*F*7$$!+FmV-qF*$!+)*fxm ))F*7$$!+ZgH^mF*$!+p#3J9)F*7$$!+9.r(G'F*$!+;jicn$F*$!+_:&* eQF*7$$!+6dtRLF*$!+plkuMF*7$$!+JmGSHF*$!+0TcIIF*7$$!+k3!>g#F*$!+TEdjEF *7$$!+uxF3AF*$!+BFXXAF*7$$!+UY&)f=F*$!+FW$=)=F*7$$!+W#zvZ\"F*$!+\"y*\\ )[\"F*7$$!+lEc86F*$!+ee?=6F*7$$!*`&\\PtF*$!+kMr]t!#67$$!*^D'\\QF*$!+^ \"H:&QFhu7$$!(\\'[()F*$!+K7l[()!#87$$\"*h=.#QF*$\"+v!z@#QFhu7$$\"*Wu?A (F*$\"+xonMsFhu7$$\"+$R1'*3\"F*$\"+!paR4\"F*7$$\"+fa;p9F*$\"+'f)*)z9F* 7$$\"+\\**[S=F*$\"+H#z<'=F*7$$\"+*\\l(*>#F*$\"+Y?]OAF*7$$\"+H(z')f#F*$ \"+'*\\6gEF*7$$\"+4>7dHF*$\"+n\"H!\\IF*7$$\"+\\a$)RLF*$\"+1\"fZZ$F*7$$ \"+*zImo$F*$\"+*p(orQF*7$$\"+4(pd1%F*$\"+&RH,K%F*7$$\"+z)4DU%F*$\"+&oS &eZF*7$$\"+\\hS&z%F*$\"+q]HP_F*7$$\"+>/)*f^F*$\"+j'*zHdF*7$$\"+zelTbF* $\"+A@8xiF*7$$\"+>ZD4fF*$\"+=0UUoF*7$$\"+Hw<&G'F*$\"+N*\\(puF*7$$\"+*p ()zl'F*$\"+8#4h:)F*7$$\"+R9c+qF*$\"++9mi))F*7$$\"+*f*=$R(F*$\"+vS,+)*F *7$$\"+44OWxF*$\"+rV^\"3\"F-7$$\"+zFy=\")F*$\"+DWDA7F-7$$\"+Cv'zH)F*$ \"+Q]H68F-7$$\"+pA:x%)F*$\"+AC'yU\"F-7$$\"+6:Vt&)F*$\"+'fS6^\"F-7$$\"+ a2rp')F*$\"+IX(Gi\"F-7$$\"+v.&yr)F*$\"+@di)p\"F-7$$\"+(***)fw)F*$\"+(f K5!=F-7$$\"+2)f+z)F*$\"+&)p\"4(=F-7$$\"+='HT\"))F*$\"+AA#f'>F-7$$\"+BX ;E))F*$\"+fj2J?F-7$$\"+H%*>Q))F*$\"+s?.?@F-7$$\"+MVB]))F*$\"+Y$p^E#F-7 $$\"+T#pA'))F*$\"+$pRR+%F--%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!FfalFe al-%*THICKNESSG6#\"\"\"-F$6%7ao7$F($!+F61uUF-F.7$F4$!+@iM>@F-7$F9$!+>_ OI?F-7$F>$!+B9>l>F-FBFGFLFQFVFenFjnF_oFdoFioF^pFcpFhp7$F^q$!+@+JYoF*Fb qFgqF\\rFarFfrF[sF`sFesFjsF_tFdtFitF^uFcuFiuF^vFdvFivF^wFcwFhwF]xFbxFg xF\\yFayFfyF[zF`zFezFjzF_[lFd[l7$Fj[l$\"+7#4h:)F*7$F_\\l$\"+*RhE'))F*F c\\lFh\\lF]]lFb]lFg]lF\\^l7$Fb^l$\"+HX(Gi\"F-Ff^lF[_lF`_lFe_lFj_l7$F`` l$\"+r?.?@F-7$Fe`l$\"+X$p^E#F-7$Fj`l$\"+F61uUF--F_al6&FaalFealFbalFeal -Fhal6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fhdl-%%VIEWG6$;$!+b#pA'))F*$\"+b# pA'))F*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 " . . . and numerically . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "xx : = 0.6;\nfn(xx);\ngn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\" \"'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%[;)))p!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%[;)))p!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "xx := 0.886;\nfn(xx); \ngn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"$'))!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+u(p_e#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+x(p_e#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "xx := 0.886225;\nfn(xx);\ngn (xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"'Di))!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*zF$\\L!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+R\"G$\\L!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'RootOfG6#,&*(-%$erfG6 #%#_ZG\"\"\"%#PiG#F1\"\"#-%$expG6#*$)-F)6#*&F-F1F2F3F4F1F1F1*&F4F1F'F1 !\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 8 \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 269 1 "y" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+``(dy/dx)^2 = 0;" "6#/,&*(%\"dG\"\"#% \"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*$-%!G6#*&%#dyGF)%#dxGF-F'F)\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 3;" "6#/-%\"yG6#\"\"!\"\"$" } {TEXT -1 2 ", " }{XPPEDIT 18 0 " `y '`(0) = 2" "6#/-%$y~'G6#\"\"!\"\"# " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "de := y(x)*diff(y(x),x$2)+diff(y(x),x)^2=0; \nic := y(0)=3,D(y)(0)=2;\ndesolve(\{de,ic\},y(x),method=missingvar,in fo=true);\nmap(expand,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,& *&-%\"yG6#%\"xG\"\"\"-%%diffG6$F(-%\"$G6$F+\"\"#F,F,*$)-F.6$F(F+F3F,F, \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"$/- -%\"DG6#F(F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,$*&-F%6$ F'F*F.F'!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_pG6#%\"yG-%%diffG6$-F(6#% \"xGF.%&~and~G/*&F%\"\"\"-F*6$F%F(F2-F*6$F,-%\"$G6$F.\"\"#%+gives~.~.~ G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\"\"-%%diffG6$F %F(F),$*&F%\"\"#F(!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&*&\"#7\"\"\"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 9 " }}{PARA 257 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = y/(2*``(dy/dx)^2);" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&F'F(*&F&F(*$-%!G6#* &%#dyGF(%#dxGF,F&F(F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6 #/-%\"yG6#\"\"!F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`y '`(0) = 1;" "6# /-%$y~'G6#\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "de := diff(y(x),x$2)=y( x)/(2*diff(y(x),x)^2);\nic := y(0)=0,D(y)(0)=1;\ndesolve(\{de,ic\},y(x ),method=missingvar,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#,$*&#\"\"\"F0F4*&F)F4-F'6$F) F,!\"#F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F* /--%\"DG6#F(F)\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,$*&# \"\"\"F.F2*&F'F2-F%6$F'F*!\"#F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_pG6#%\" yG-%%diffG6$-F(6#%\"xGF.%&~and~G/*&F%\"\"\"-F*6$F%F(F2-F*6$F,-%\"$G6$F .\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"y G\"\"\"-%%diffG6$F%F(F),$*&#F)\"\"#F)*&F(F)F%!\"#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 98 "fn := proc(x)\n local y;\n fsolve(y*hypergeom([1/ 2,1/4],[3/2],-y^2)=x,y=x*(x^2/40+1));\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can also construct a n umerical solution " }{TEXT 262 2 "gn" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "de := \+ diff(y(x),x$2)=y(x)/(2*diff(y(x),x)^2);\nic := y(0)=0,D(y)(0)=1;\ngn : = desolve(\{de,ic\},y(x),x=-10..10,type=numeric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#,$*&#\"\" \"F0F4*&F)F4-F'6$F)F,!\"#F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ic G6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The two solutions can be compared \+ graphically . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 69 "plot(['fn'(x),'gn'(x)],x=-10..10,color=[red, green],thickness=[1,2]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7S7$$!#5\"\"!$!+6qUMJ!\")7$$!+s%HaF1$!+!=9m4\"F-7$$!+]$*4)*\\F1$!+VL*4e*F1 7$$!+]_&\\c%F1$!+zBE$G)F17$$!+]1aZTF1$!+&zt27(F17$$!+/#)[oPF1$!+'Rl)Rh F17$$!+$=exJ$F1$!+%e=d1&F17$$!+L2$f$HF1$!+lPvLUF17$$!+PYx\"\\#F1$!+6>: bLF17$$!+L7i)4#F1$!++QIcEF17$$!+P'psm\"F1$!+a*\\D(>F17$$!+74_c7F1$!+e' *R)R\"F17$$!*3x%z#)F1$!+,20=()F)7$$!*?PQM%F1$!+5rg5WF)7$$!(\"zr)*F1$!+ o6(=()*!#77$$\")!o2J%F-$\"+5i/wVF)7$$\")%Q#\\\")F-$\"+B#=$o&)F)7$$\"*; *[H7F-$\"+RY5j8F17$$\"*qvxl\"F-$\"+U8Xe>F17$$\"*`qn2#F-$\"+0Ig>EF17$$ \"*cp@[#F-$\"+v>?PLF17$$\"*3'HKHF-$\"+#4whA%F17$$\"*xanL$F-$\"+%ft*3^F 17$$\"*v+'oPF-$\"+1h9ShF17$$\"*S<*fTF-$\"+]/+arF17$$\"*&)Hxe%F-$\"+^a< \\$)F17$$\"*.o-*\\F-$\"+=Cqc&*F17$$\"*TO5T&F-$\"+]\\]!4\"F-7$$\"*U9C#e F-$\"+KE&3B\"F-7$$\"*1*3`iF-$\"+q+#oQ\"F-7$$\"*$*zym'F-$\"+X'zda\"F-7$ $\"*^j?4(F-$\"+Mr@<#z&*GF-7$$\"#5F*$\"+6qUMJF--%'COLOURG6&%$RGB G$\"*++++\"F-$F*F*F`[l-%*THICKNESSG6#\"\"\"-F$6%7SF'F.F4F9F>FCFHFMFRFW FfnF[o7$Fao$!+WL*4e*F1FeoFjo7$F`p$!+(Rl)RhF17$Fep$!+&e=d1&F17$Fjp$!+mP vLUF1F^qFcqFhqF]rFbrFgrF\\sFbsFgsF\\tFatFftF[uF`uFeuFjuF_vFdvFivF^wFcw FhwF]xFbxFgxF\\yFayFfyF[zF`zFez-F[[l6&F][lF`[lF^[lF`[l-Fb[l6#\"\"#-%+A XESLABELSG6$Q\"x6\"Q!F]]l-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " . . . and nume rically . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "xx := 3;\nfn(xx);\ngn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+JRQo V!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+JRQoV!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "xx := 1 0;\nfn(xx);\ngn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6qUMJ!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6qUMJ!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "A numerical solution constructed by " } {TEXT 0 6 "dsolve" }{TEXT -1 47 " does not require an interval to be s pecified. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 136 "de := diff(y(x),x$2)=y(x)/(2*diff(y(x),x)^2);\nic \+ := y(0)=0,D(y)(0)=1;\nhn := dsolve(\{de,ic\},y(x),type=numeric,abserr= 1e-10,relerr=1e-10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diff G6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#,$*&#\"\"\"F0F4*&F)F4-F'6$F)F,!\"#F4F4 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6# F(F)\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "xx := 1000;\nfn(xx);\nevalf(subs(hn(xx),y(x)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"%+5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!H%*f]#!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ !H%*f]#!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'RootOfG6#,&*&%#_ZG\"\"\"-%*hypergeomG6 %7$#F.\"\"%#F.\"\"#7##\"\"$F6,$*$)F-F6F.!\"\"F.F=F'F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 10 " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = y^2/(``(dy/dx)^2);" "6 #/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&F'F&*$-%!G6#*&%#dyGF( %#dxGF,F&F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6# \"\"!F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 1;" "6#/-%$y~'G6 #\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "de := diff(y(x),x$2)=y(x)^2 /diff(y(x),x)^2;\nic := y(0)=0,D(y)(0)=1;\ndesolve(\{de,ic\},y(x),meth od=missingvar,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-% %diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&F)F0-F'6$F)F,!\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#*&F'F.-F%6$F'F*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Sub stituting~.~.~G/-%#_pG6#%\"yG-%%diffG6$-F(6#%\"xGF.%&~and~G/*&F%\"\"\" -F*6$F%F(F2-F*6$F,-%\"$G6$F.\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\"\"-%%diffG6$F%F(F)*&F(\"\"#F%!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 96 "plots[implic itplot](y*hypergeom([1/4,1/3],[4/3],-4/3*y^3)=x,\n x=-0.6..5,y=-0.6. .11,color=red);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6fp7$7$$!3w**************f!#=$!3]hLD/%4>)eF*7$$!3G!yZuJya ^%F*$!3g#)Q9G13NWF*7$7$$!3-++++++gPF*$!33f)eOO%y)p$F*F-7$F37$$!3]!)yHX !)=#*HF*$!3;\"Ra/in/&HF*7$7$$!3_++++++?:F*$!3pcV1B$fc^\"F*F97$7$$!3C++ ++++?:F*$!3TcV1B$fc^\"F*7$$!3#*zz9tx*)o9F*$!39**[w7Y&eY\"F*7$7$$!3K+++ FaGg8F*$!35++++++g8F*FJ7$7$FQ$!3Q++++++g8F*7$$\"31'3#fJ(p!zT!#?$\"3Ig[ %))p7i[%Ffn7$7$$\"3I&************>(!#>$\"3d5E.!p!>WsF]oFY7$7$F[o$\"3>4 E.!p!>WsF]o7$$\"3+l7Yg;^^:F*$\"39@X/*)Hed:F*7$7$$\"3I************fHF*$ \"3_r9os+()oHF*Fdo7$Fjo7$$\"3)*3mgBEBhIF*$\"3o$f+6&QIqIF*7$7$$\"3!*)** **4#H^qKF*$\"3e************zKF*F`p7$7$$\"3Y*****4#H^qKF*$\"39++++++!G$ F*7$$\"3Q$y>B_8ra%F*$\"3^hhw.xSKYF*7$7$$\"31*************>&F*$\"3%Hn'4 *)Q1C`F*Faq7$Fgq7$$\"3i%HyC$y:HgF*$\"3rt1ev(eC?'F*7$7$$\"3%))********* **RuF*$\"3g&R:)y&zqp(F*F]r7$Fcr7$$\"3'e!ojU@?6vF*$\"3!f=&RZ)4Dx(F*7$7$ $\"3#*)****\\'\\U]wF*$\"3:)***********>zF*Fir7$F_s7$$\"3V,rMOI)p\"*)F* 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3l*****>7\"HWCF`t$\"3:************RSF`tFj^l7$F`_l7$$\"3=k&yzFP.[#F`t$ \"3s(oeFN,`:%F`t7$7$$\"3f***********f`#F`t$\"3C:(R[\\]LL%F`tFf_l7$F\\` l7$$\"3#=([uK/QoDF`t$\"3.]qXYi#pV%F`t7$7$$\"3u*****fS\\$*e#F`t$\"3c*** ********R]%F`tFb`l7$Fh`l7$$\"3;h38RNK`EF`t$\"3x9K%*oE(\\s%F`t7$7$$\"3o *****4v#oBFF`t$\"3)************z'\\F`tF^al7$Fdal7$$\"3c!f\">$>4nt#F`t$ \"3%=cuECXi,&F`t7$7$$\"3!)************fFF`t$\"3p#*z3*)f]-^F`tFjal7$7$F abl$\"3e$*z3*)f]-^F`t7$$\"3a;C()pX0$F`t$\"3mE5$ffZ4'oF`t7$7$$\"3=++++++3KF`t$\"3;Q-&Ql^R%p F`tF[gl7$7$$\"3j++++++3KF`tFdgl7$$\"3_`A(H()f\"fKF`t$\"3y>YTxf-#=(F`t7 $7$$\"3)******pmK>G$F`t$\"3/-+++++)G(F`tFjgl7$7$$\"3U+++nE$>G$F`tFchl7 $$\"3k#)Ri\")[kELF`t$\"3FE.N-cB1vF`t7$7$$\"3m+++4s*pP$F`t$\"3W-+++++_x F`tFihl7$7$$\"3A+++4s*pP$F`tFbil7$$\"3O()oL2Q(GR$F`t$\"3r$ftL6ZI$yF`t7 $7$$\"3Q++++++KMF`t$\"362%)3G\"zF.)F`tFhil7$F^jl7$$\"35MNH\"eJvX$F`t$ \"3)[#[.`J6j\")F`t7$7$$\"3K+++o<*yY$F`t$\"3'G+++++g@)F`tFdjl7$Fjjl7$$ \"3c7V\"G7p0_$F`t$\"3gqgX)QNl\\)F`t7$7$$\"3]+++u!>]b$F`t$\"3E.+++++!o) F`tF`[m7$Ff[m7$$\"3_eX10*)[#e$F`t$\"3*oFmB)HFK))F`t7$7$$\"3`+++>SsQOF` t$\"3!>+++++S9*F`tF\\\\m7$7$Fc\\m$\"3o.+++++W\"*F`t7$$\"3'QQ!o'e%HVOF` t$\"3#p1ihN=.<*F`t7$7$$\"3e++++++cOF`t$\"3Ky)=A)*yMC*F`tF[]m7$Fa]m7$$ \"3vf*H%pkb-PF`t$\"3'\\%z.*fh:^*F`t7$7$$\"3q+++4WI>PF`t$\"34/+++++3'*F `tFg]m7$7$$\"39,++4WI>PF`tF`^m7$$\"3!)GruNfpgPF`t$\"3w_4QZ)H^&)*F`t7$7 $$\"3G+++(H>qz$F`t$\"3X+++++?25!#;Ff^m7$F\\_m7$$\"3Xycw.(\\y\"QF`t$\"3 )HDM**)R2?5Fa_m7$7$$\"31,++Yb4sQF`t$\"3o+++++g`5Fa_mFc_m7$7$$\"3g+++Yb 4sQF`t$\"3]+++++g`5Fa_m7$$\"3o?!\\h5JS(QF`t$\"3#[FEPTO[0\"Fa_m7$7$$\"3 y++++++!)QF`t$\"3IYc(\\F\\'e5Fa_mFd`m7$Fj`m7$$\"3R%*)3p)=))GRF`t$\"3Ce JGmW()*3\"Fa_m7$7$$\"3m+++PItWRF`t$\"3a+++++++6Fa_mF`am-%+AXESLABELSG6 $%\"xG%\"yG-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!FhbmFgbm" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The solution is the \+ inverse of the function " }{XPPEDIT 18 0 "f(y)=y*hypergeom([1/4, 1/3] ,[4/3],-4/3*y^3)" "6#/-%\"fG6#%\"yG*&F'\"\"\"-%*hypergeomG6%7$*&F)F)\" \"%!\"\"*&F)F)\"\"$F07#*&F/F)F2F0,$*(F/F)F2F0F'F2F0F)" }{TEXT -1 3 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 22 "A numerical procedure " }{TEXT 262 2 "fn" }{TEXT -1 39 " to evaluate the inverse is as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "fn := proc(x)\n local y;\n fsolve(y*hypergeom([1/4,1/3],[4/3],-4/3*y ^3)=x,y=x*(x-1));\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 43 "We can also construct a numerical solutio n " }{TEXT 262 2 "gn" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "de := diff(y(x),x$2)=y( x)^2/diff(y(x),x)^2;\nic := y(0)=0,D(y)(0)=1;\ngn := desolve(\{de,ic\} ,y(x),x=-0.9823..10,type=numeric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&F)F0-F'6$F)F,!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F) \"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The two solutions can be compared graphically . . . " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot( ['fn'(x),'gn'(x)],x=-0.9823..5,color=[red,green],thickness=[1,2]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7S 7$$!%B)*!\"%$!++(*4t))!#57$$!+Y)G!>&)F-$!+taSH!)F-7$$!+=0X%Q(F-$!+C[$* >rF-7$$!+D&)\\3hF-$!+-%f$))fF-7$$!+&p)3C[F-$!+,O?yZF-7$$!+SMyXNF-$!+Wd _KNF-7$$!+N3jgBF-$!+Z\"Q!eBF-7$$!+fRZL6F-$!+zjLL6F-7$$\"*QMlN\"F-$\"+i Y`c8!#67$$\"*t52S\"!\"*$\"+)QJ5S\"F-7$$\"*kx>q#FZ$\"+toS1FF-7$$\"*DM\" [QFZ$\"+u&F-7$$\"*5MSV'FZ$\"+Fn&=d'F-7$$\" *y$e#o(FZ$\"+@7aczF-7$$\"*^'Q;))FZ$\"+a\"y&z#*F-7$$\"+h(ek,\"FZ$\"+N(4 a4\"FZ7$$\"+z!p18\"FZ$\"+IFXZ7FZ7$$\"+HG_j7FZ$\"+@<(yV\"FZ7$$\"+547\"Q \"FZ$\"+[Z-?;FZ7$$\"+UY95:FZ$\"+2E3O=FZ7$$\"+Zd+L;FZ$\"+hF%*e?FZ7$$\"+ @%)>h*y=FZ$\"+Y#=.c#FZ7$$\"+*>(*e+#FZ$\"+,7o]G FZ7$$\"+O:zP@FZ$\"+iK(o<$FZ7$$\"+Wfg_AFZ$\"+CAQ#[$FZ7$$\"+]'3mP#FZ$\"+ l94OQFZ7$$\"+zbr/DFZ$\"+f]tGUFZ7$$\"+;K/IEFZ$\"+V`*4k%FZ7$$\"+\\TI^FFZ $\"+S`ln]FZ7$$\"+SP%f)GFZ$\"+IT FZ$\"+U)=aA\"F]x7$$\"+>;-cUFZ$\"+^YT;8F]x7$$\"+SakrVFZ$\"+1<+([FZ$\"+xZ9N=F]x7$$\"\"&\"\"!$\"+]8;i>F]x-%'COLOURG6& %$RGBG$\"*++++\"F]x$FizFizFb[l-%*THICKNESSG6#\"\"\"-F$6%7SF'F.F3F8F=FB FGFLFQFWFgn7$F]o$\"+t&F-FfoF[p7$Fap$\"+`\"y&z#*F-F epFjpF_qFdqFiqF^rFcrFhrF]sFbs7$Fhs$\"+DAQ#[$FZF\\tFatFftF[uF`uFeu7$F[v $\"+*\\&)ph'FZF_v7$Fev$\"+JZbsxFZ7$Fjv$\"+Tmz#R)FZ7$F_w$\"+T!4+3*FZFcw FhwF^xFcxFhxF]yFbyFgyF\\zFaz7$Fgz$\"+^8;i>F]x-F][l6&F_[lFb[lF`[lFb[l-F d[l6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!F^^l-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " \+ . . . and numerically . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "xx := 3;\nfn(xx);\ngn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+B;`Lg!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+B; `Lg!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "xx := 10;\nfn(xx);\ngn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+`^A89 !\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+`^A89!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "xx := - 0.9823;\nfn(xx);\ngn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!% B)*!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!++(*4t))!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!++(*4t))!#5" }}}{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-%'RootOfG6#*&)\"\"$#\" \"\"\"\"%F/,&*&%#_ZGF/-%*hypergeomG6%7$F.#F/F-7##F0F-,$*(F0F/F-!\"\"F3 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 11 "Exa mple 11 " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d *x^2) = y^2*``(dy/dx);" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(! \"\"*&F'F&-%!G6#*&%#dyGF(%#dxGF,F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y ' `(0) = 1;" "6#/-%$y~'G6#\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "de := dif f(y(x),x$2)=y(x)^2*diff(y(x),x);\nic := y(0)=0,D(y)(0)=1;\ndesolve(\{d e,ic\},y(x),method=missingvar,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&)F)F0\" \"\"-F'6$F)F,F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\" \"!F*/--%\"DG6#F(F)\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\" #*&)F'F.\"\"\"-F%6$F'F*F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_pG6#%\"yG-%%d iffG6$-F(6#%\"xGF.%&~and~G/*&F%\"\"\"-F*6$F%F(F2-F*6$F,-%\"$G6$F.\"\"# %+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\" \"-%%diffG6$F%F(F)*&)F(\"\"#F)F%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%F'F)F)#F)\"\"'F)F@F)F)F)*&F(F)*&)F'#\"\"&FEF)-%'ar ctanG6#*&,&*&#F>\"\"*F)*&FBF)F1F)F)F)#F)F'F@F)F'#F)F>F)F)F)*(\"#=F@F'F I%#PiGF)F)F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "We can construct a numerical solution " }{TEXT 262 2 "gn " }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 132 "de := diff(y(x),x$2)=y(x)^2*diff(y(x),x);\n ic := y(0)=0,D(y)(0)=1;\ngn := desolve(\{de,ic\},y(x),x=-3..1.7439,typ e=numeric,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%d iffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&)F)F0\"\"\"-F'6$F)F,F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The \+ two solutions can be compared graphically . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "f := y -> ( -1/6*ln(y^2-y*3^(1/3)+3^(2/3))+1/3*ln(y+3^(1/3)))*3^(1/3)\n +1/3*3^( 5/6)*arctan((2/9*3^(2/3)*y-1/3)*sqrt(3))+1/18*3^(5/6)*Pi;\np1 := plots [implicitplot](f(y)=x,x=-3..0,y=-1.5..0,color=red):\np2 := plots[impli citplot](f(y)=x,x=0..1.7439,y=0..20,color=red):\np3 := plot('gn'(x),x= -3..1.7439,y=-1.5..20,color=green,thickness=2):\nplots[display]([p1,p2 ,p3]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"yG6\"6$%)oper atorG%&arrowGF(,(*&,&*&#\"\"\"\"\"'F1-%#lnG6#,(*$)9$\"\"#F1F1*&F9F1)\" \"$#F1F=F1!\"\"*$)F=#F:F=F1F1F1F?*&F>F1-F46#,&F9F1*$FF1*&)F=#\"\"&F2F1-%'arctanG6#*&,&*&#F:\"\"*F1*&FAF1F9F1F1F1#F1F=F? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "de := diff( y(x),x$2)=sin(y(x))*diff(y(x),x);\nic := y(0)=0,D(y)(0)=1;\ndesolve(\{ de,ic\},y(x),method=missingvar,info=true);\ng := unapply(rhs(%),x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6 $F,\"\"#*&-%$sinG6#F)\"\"\"-F'6$F)F,F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-% \"yG6#%\"xG-%\"$G6$F*\"\"#*&-%$sinG6#F'\"\"\"-F%6$F'F*F3" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substit uting~.~.~G/-%#_pG6#%\"yG-%%diffG6$-F(6#%\"xGF.%&~and~G/*&F%\"\"\"-F*6 $F%F(F2-F*6$F,-%\"$G6$F.\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\"\"-%%diffG6$F%F(F)*&-%$sinGF'F)F%F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%\"gGf*6#%\"xG6\"6$%)operatorG %&arrowGF(,$*&\"\"#\"\"\"-%'arctanG6#,$*&#F/\"\"$F/*&-%$tanG6#,$*&#F/F .F/*&9$F/F6F=F/F/F/F6F=F/F/F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "We ca n construct a numerical solution " }{TEXT 262 2 "gn" }{TEXT -1 3 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "de := diff(y(x),x$2)=sin(y(x))*diff(y(x),x);\nic := y(0)=0,D( y)(0)=1;\ngn := desolve(\{de,ic\},y(x),x=-8..8,type=numeric,method=rk7 8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG- %\"$G6$F,\"\"#*&-%$sinG6#F)\"\"\"-F'6$F)F,F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The two solutio ns can be compared graphically . . . 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "de := diff(y(x),x$2)=arctan(y(x))*diff(y(x),x);\nic \+ := y(0)=0,D(y)(0)=1;\ndesolve(\{de,ic\},y(x),method=missingvar,info=tr ue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG -%\"$G6$F,\"\"#*&-%'arctanG6#F)\"\"\"-F'6$F)F,F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%d iffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#*&-%'arctanG6#F'\"\"\"-F%6$F'F*F3" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6'%2Substituting~.~.~G/-%#_pG6#%\"yG-%%diffG6$-F(6#%\"xGF.%&~and~G/*&F %\"\"\"-F*6$F%F(F2-F*6$F,-%\"$G6$F.\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\"\"-%%diffG6$F%F(F)*&-%'arctanG F'F)F%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The solution is the inverse of \+ the function " }{XPPEDIT 18 0 "f(y) = Int(2/(2*t*arctan(t)-ln(t^2+1)+ 2),t = 0 .. y);" "6#/-%\"fG6#%\"yG-%$IntG6$*&\"\"#\"\"\",(*(F,F-%\"tGF --%'arctanG6#F0F-F--%#lnG6#,&*$F0F,F-F-F-!\"\"F,F-F9/F0;\"\"!F'" } {TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "We can construct a numerical solution " }{TEXT 262 2 "gn " }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 133 "de := diff(y(x),x$2)=arctan(y(x))*diff(y(x) ,x);\nic := y(0)=0,D(y)(0)=1;\ngn := desolve(\{de,ic\},y(x),x=-8..8,ty pe=numeric,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%% diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&-%'arctanG6#F)\"\"\"-F'6$F)F,F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F( F)\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The two solutions can be compared graphically . . . " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 235 "f := y -> Int(2/(2*t*arctan(t)-ln(t^2+1)+2),t=0..y);\np1 := plots[implicit plot](Int(2/(2*t*arctan(t)-ln(t^2+1)+2),t=0..y)=x,\n x=-3..3,y=-15.. 15,color=red):\np2 := plot('gn'(x),x=-3..3,color=green,thickness=2):\n plots[display]([p1,p2]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf* 6#%\"yG6\"6$%)operatorG%&arrowGF(-%$IntG6$,$*&\"\"#\"\"\",(*(F1F2%\"tG F2-%'arctanG6#F5F2F2-%#lnG6#,&*$)F5F1F2F2F2F2!\"\"F1F2F?F2/F5;\"\"!9$F (F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURV ESG6_q7$7$$!\"$\"\"!$!3S'>oq7$7 $$!3E+++6#e4)GF1FG7$$!3S*3&*pmFE&GF1$!3UbC]mho87F-7$7$$!3(******pC`v!G F1$!3A++++++S6F-FM7$FS7$$!3M=%\\HK,az#F1$!3%4HD&Q$*HA6F-7$7$$!3!)***** *******fFF1$!3-C`YzKtq5F-FY7$Fin7$$!3f#4H&y32MFF1$!36aatgX'H.\"F-7$7$$ !3m*******Rq^s#F1$!3H++++++?5F-F_o7$Feo7$$!3E`**\\.r_nEF1$!3UM-]#[kBY* F17$7$$!3#)******[^RJEF1$!3c.++++++!*F1F[p7$Fap7$$!3g%y+ERvmf#F1$!3X\" 3'*p.Bmh)F17$7$$!3\\*****\\e>F_#F1$!3[-++++++yF1Fgp7$7$F^q$!3Q.++++++y F17$$!3'Q\"\\oc\"p<_#F1$!3GHad;U:\"z(F17$7$$!3d************>DF1$!3]R(* \\,%)ouxF1Ffq7$F\\r7$$!3\\$G&zP]*yV#F1$!3g$eB5\"[_5qF17$7$$!3#)******= ly$R#F1$!3?.++++++mF1Fbr7$7$$!3Q******=ly$R#F1F[s7$$!3A9vI>#H'[BF1$!3w HCY.R&oD'F17$7$$!3O************zAF1$!3S*yN_)oMNdF1Fas7$Fgs7$$!3WL]xdGQ `AF1$!3sJ[76d3LbF17$7$$!3I******f!peB#F1$!3+.++++++aF1F]t7$Fct7$$!3OO_ IC*ei9#F1$!3!z\"QZy`qo[F17$7$$!3:************R?F1$!3ubv%)33qQUF1Fit7$F _u7$$!3oO.Qu)ek.#F1$!3&pJ)4GcqF1$!3Apt+v%o*eOF17$7$$!3Q*************z\"F1$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "de := diff(y(x),x$2)=diff(y (x),x)*(diff(y(x),x)+y(x));\nic := y(0)=0,D(y)(0)=1;\ndesolve(\{de,ic \},y(x),method=missingvar,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&-F'6$F)F,\"\"\",&F2F 4F)F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/-- %\"DG6#F(F)\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,&*$)-F%6 $F'F*F.\"\"\"F4*&F2F4F'F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_pG6#%\"yG-%%d iffG6$-F(6#%\"xGF.%&~and~G/*&F%\"\"\"-F*6$F%F(F2-F*6$F,-%\"$G6$F.\"\"# %+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\" \"-%%diffG6$F%F(F),&*$)F%\"\"#F)F)*&F%F)F(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 143 "de := diff(y(x),x$2)=diff(y(x),x)*(diff(y(x ),x)+y(x));\nic := y(0)=0,D(y)(0)=1;\ngn := desolve(\{de,ic\},y(x),x=- 3..0.83,type=numeric,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&-F'6$F)F,\"\"\",&F2F4F) F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\" DG6#F(F)\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The two solutions can be compared graphically . . . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "h := y -> Int(-1/(t+1-2*exp(t)),t=0..y);\np1 := plots[implicitplo t](h(y)=x,x=-3..0.83,y=-3.5..8,color=red,grid=[35,35]):\np2 := plot('g n'(x),x=-3..0.83,color=green,thickness=2):\nplots[display]([p1,p2]);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"yG6\"6$%)operatorG%&a rrowGF(-%$IntG6$,$*&\"\"\"F1,(%\"tGF1F1F1*&\"\"#F1-%$expG6#F3F1!\"\"F9 F9/F3;\"\"!9$F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6as7$7$$!\"$\"\"!$!3I?8:%RD6,$!#<7$$!3i2P#*y7 ;jHF-$!3A6Rkf<9MHF-7$7$$!3$4jS@KI-\"HF-$!3i0Zw6%HN#GF-F.7$7$$!3QJ19A.B 5HF-F77$$!3i]dZiH!f*GF-$!33j90Cq&yz#F-7$7$$!3Yqko#F-$!3ue>jXgfDCF-7$7$$ !3O6%HN#)e?m#F-$!3coE]!*oc&R#F-F^o7$Fdo7$$!3^_6C!f$*pg#F-$!3D[#\\oH)R7 BF-7$7$$!3#=)eqk#F-7$7$$!3aory,P_(\\#F-$!3C6%HN#)eq9#F-Ffp7$F\\q7$$!3Gi'ocDWYX#F- $!33mGF\"\\tL4#F-7$7$$!3G_B)eqknV#F-$!3EoS9%=))42#F-Fbq7$Fhq7$$!3eo*ep %e9vBF-$!3gi5$4*4%Q*>F-7$7$$!3tA)eqkF-F^r7$Fdr7$$!3 Uu#\\#Quk&H#F-$!33f#*e!\\3V*=F-7$7$$!39(Q5K0ptA#F-$!30k$HN#)eq9@#F-$!3%[**Hh#)3Dz \"F-Ffs7$F\\t7$$!39&)\\s'fl:8#F-$!3f%pX!R=^5F-$!3+$)[vuiJh:F-F^u7$Fdu7$$!3eWe1fllj>F-$!3y()eeon?Q:F-7$7$$!3q Sns#=jx*=F-$!3'o6%HN#)eq9F-Fju7$7$$!3#4uEF=jx*=F-Fcv7$$!33(4 z\"F-$!3Eexri3C!Q\"F-7$7$$!3ov6%HN#)3w\"F-$!3MwRJ'\\%ya8F-Few7$F[x7$$! 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" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` (d^2*y/(d*x^2))/((1+(dy/dx)^2)^(3/2)) = 1;" "6#/*&-%!G6#*(%\"dG\"\"#% \"yG\"\"\"*&F)F,*$%\"xGF*F,!\"\"F,),&F,F,*$*&%#dyGF,%#dxGF0F*F,*&\"\"$ F,F*F0F0F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6# \"\"!\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 0;" "6#/-%$y ~'G6#\"\"!F'" }{TEXT -1 2 ". 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\"\"\"/--%\"DG6#F(F)F*" }}{PARA 7 "" 1 "" {TEXT -1 42 "Warning, '_Choo seSolutionDE1' is set to 1\n" }}{PARA 7 "" 1 "" {TEXT -1 42 "Warning, \+ '_ChooseSolutionDE2' is set to 2\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"yG6#%\"xG,&\"\"#\"\"\"*$,&*$)F'F)F*!\"\"F*F*#F*F)F/" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"# \"\"\"*$,&*$)9$F-F.!\"\"F.F.#F.F-F4F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 293 222 222 {PLOTDATA 2 "6%-%'CURVESG6$7Y7$$\"\"!F)$\"\"\"F) 7$$\"3emmm;arz@!#>$\"38\"[)='eP-+\"!#<7$$\"3[LL$e9ui2%F/$\"3U#RZf9J3+ \"F27$$\"3nmmm\"z_\"4iF/$\"3S]F[S&H>+\"F27$$\"3[mmmT&phN)F/$\"3\"z3aVR (\\.5F27$$\"3CLLe*=)H\\5!#=$\"3&\\k*pq._05F27$$\"3gmm\"z/3uC\"FE$\"3M. p@P1\"y+\"F27$$\"3%)***\\7LRDX\"FE$\"3J2K$=f01,\"F27$$\"3]mm\"zR'ok;FE $\"3aa%=1D`R,\"F27$$\"3w***\\i5`h(=FE$\"3sO$*G:uv<5F27$$\"3WLLL3En$4#F E$\"3lz<_AH;A5F27$$\"3qmm;/RE&G#FE$\"3#pwN2Flmu0\"F27$$\"3,LLLLY.KNFE$\"3q$fD!oMXk5F27$$\" 3w***\\7o7Tv$FE$\"3i_kVh;9t5F27$$\"3'GLLLQ*o]RFE$\"3![z8U_[83\"F27$$\" 3A++D\"=lj;%FE$\"3aBs:Ro#44\"F27$$\"31++vV&RY2aFE$\"3MW@_'>9)e6F27$$\"39mm;zXu9cFE$\"3a\\&y;$f]s6F27$ $\"3l******\\y))GeFE$\"3#=#fQa![u=\"F27$$\"3'*)***\\i_QQgFE$\"3S`YtNL* G?\"F27$$\"3@***\\7y%3TiFE$\"3Wl?5'Hi'=7F27$$\"35****\\P![hY'FE$\"3e&Q pXJ$=P7F27$$\"3kKLL$Qx$omFE$\"3%3.&*R0(za7F27$$\"3!)*****\\P+V)oFE$\"3 e)pGqC(pu7F27$$\"3?mm\"zpe*zqFE$\"3EABtEMy$H\"F27$$\"3%)*****\\#\\'QH( FE$\"3)>'e_Z\")*eJ\"F27$$\"3GKLe9S8&\\(FE$\"3lgI7I3,Q8F27$$\"3R***\\i? =bq(FE$\"3G3PtXEii8F27$$\"3\"HLL$3s?6zFE$\"35$Qi&f*R$)Q\"F27$$\"3a*** \\7`Wl7)FE$\"3f3;)y8asT\"F27$$\"3#pmmm'*RRL)FE$\"3-L#>fm?tW\"F27$$\"3Q mm;a<.Y&)FE$\"3/Mkto$=2[\"F27$$\"3=LLe9tOc()FE$\"3@jd-'pHq^\"F27$$\"3u ******\\Qk\\*)FE$\"3!yp%e4C'Qb\"F27$$\"3CLL$3dg6<*FE$\"3W7+>2(z8g\"F27 $$\"3ImmmmxGp$*FE$\"3)4m')3mz/l\"F27$$\"3A++D\"oK0e*FE$\"3;ext7)4Mr\"F 27$$\"3C+++]oi\"o*FE$\"3s&o$Rw\"y'\\F27$F*$\"\"#F)-%'COLOURG6&%$RGBG$ \"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!F`]l-%%VIEWG6$;F(F*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Example 16 " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = (dy/dx-1)^ 2;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*$,&*&%#dyGF(%#dx GF,F(F(F,F&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6# \"\"!\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 2;" "6#/-%$y ~'G6#\"\"!\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "de := diff(y(x),x$2)=(diff( y(x),x)-1)^2;\nic := y(0)=1,D(y)(0)=2;\nsoln := desolve(\{de,ic\},y(x) ,method=missingvar,info=true);\ng := unapply(rhs(soln),x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\" #*$),&-F'6$F)F,\"\"\"F6!\"\"F0F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #icG6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"y G6#%\"xG-%\"$G6$F*\"\"#,(*$)-F%6$F'F*F.\"\"\"F4*&F.F4F2F4!\"\"F4F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 '%2Substituting~.~.~G/-%#_pG6#%\"yG-%%diffG6$-F(6#%\"xGF.%&~and~G/*&F% \"\"\"-F*6$F%F(F2-F*6$F,-%\"$G6$F.\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_pG6#%\"yG\"\"\"-%%diffG6$F%F(F),(*$)F%\"\"#F) F)*&F0F)F%F)!\"\"F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%%solnG/-%\"yG6#%\"xG,&\"\"#\"\"\"-%#lnG6#,$*&,&F,!\"\"F)F,F,-% $expG6#,&F,F,F)F3F,F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,&\"\"#\"\"\"-%#lnG6#,$*&,&9$F.F.!\"\"F. -%$expG6#,&F.F.F5F6F.F6F6F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs(soln,de);\nsimplify(%,s ymbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$,&\"\"#\"\"\"- %#lnG6#,$*&,&F)!\"\"%\"xGF)F)-%$expG6#,&F)F)F1F0F)F0F0-%\"$G6$F1F(*$), &-F%6$F'F1F)F)F0F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&\"\"\"F%*$) ,&F%!\"\"%\"xGF%\"\"#F%F)F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dsolve(\{de,ic\},y(x));\nh : = unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG ,&F'\"\"\"-%#lnG6#,&-%$expG6#!\"\"F)*&F.F)F'F)F1F1" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"-% #lnG6#,&-%$expG6#!\"\"F.*&F3F.F-F.F6F6F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([g(x),h (x)],x=-3..1,thickness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 364 327 327 {PLOTDATA 2 "6&-%'CURVESG6%7gn7$$!\"$\"\"!$!3!3*)>6O%H'Q$!#<7$$!3? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "de := diff( y(x),x$2)=diff(y(x),x)/(diff(y(x),x)+1);\nic := y(1)=0,D(y)(1)=3;\nsol := desolve(\{de,ic\},method=missingvar,info=true);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&-F'6$F )F,\"\"\",&F2F4F4F4!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/- %\"yG6#\"\"\"\"\"!/--%\"DG6#F(F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xG-%\" $G6$F*\"\"#*&-F%6$F'F*\"\"\",&F0F2F2F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~. ~.~G/-%#_pG6#%\"yG-%%diffG6$-F(6#%\"xGF.%&~and~G/*&F%\"\"\"-F*6$F%F(F2 -F*6$F,-%\"$G6$F.\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*&-%#_pG6#%\"yG\"\"\"-%%diffG6$F%F(F)*&F%F),&F%F)F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%$solG/,,\"\"$!\"\"*$,&\"#;\"\"\"*&\"\"#F,-%\"yG6#%\"x GF,F,#F,F.F,-%#lnG6#*$,&*&F.F,F/F,F,\"#:F,F3F,-F56#*$,&F,F(F)F,F3F,-F5 6#,$*&#F,F'F,*&F,F,*$,&F,F,F)F,#F,F.F(F,F,F,F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "h := unapply(lhs(subs(y(x)=y,sol)),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"yG6\"6$%)operatorG%&arrowGF(,, \"\"$!\"\"*$,&\"#;\"\"\"*&\"\"#F29$F2F2#F2F4F2-%#lnG6#*$,&*&F4F2F5F2F2 \"#:F2F6F2-F86#*$,&F2F.F/F2F6F2-F86#,$*&#F2F-F2*&F2F2*$,&F2F2F/F2#F2F4 F.F2F2F2F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 6 "dsolve" }{TEXT -1 34 " can obtain an explicit solution. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "de := diff(y(x),x$2)=diff(y(x),x)/(diff(y(x),x)+1);\nic := y(1) =0,D(y)(1)=3;\ndsolve(\{de,ic\},y(x)):\ng := unapply(rhs(%),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$ F,\"\"#*&-F'6$F)F,\"\"\",&F2F4F4F4!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"\"\"\"!/--%\"DG6#F(F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&#\"\" \"\"\"#F/*$)-%)LambertWG6#,$**\"\"$F/-%$expG6#9$F/-F:6#F8F/-F:6#F/!\" \"F/F0F/F/F/F3F/#\"#:F0FAF(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "We can compare the two solutions graphica lly . . . 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Rmr\\F\\dm7$$\"*Ey'G**F\\dm$!)[uP@F\\dm7$$\"+J%=H<\"F\\dm$\"*Sz+I&F\\d m7$$\"+3>qM8F\\dm$\"+-2SY5F\\dm7$$\"+,.W2:F\\dm$\"+cO')>;F\\dm7$$\"+fp 'Rm\"F\\dm$\"+_WRf@F\\dm7$$\"+T>4N=F\\dm$\"+]R1rFF\\dm7$$\"+8s5'*>F\\d m$\"+g>dnLF\\dm7$$\"+mXTk@F\\dm$\"+]y*H,%F\\dm7$$\"+od'*GBF\\dm$\"+!*o \"em%F\\dm7$$\"+EcB,DF\\dm$\"+Su]s`F\\dm7$$\"+v>:nEF\\dm$\"+qCwvgF\\dm 7$$\"+0a#o$GF\\dm$\"+qs/=oF\\dm7$$\"+`Q40IF\\dm$\"+qVRxvF\\dm7$$\"+\"3 :(fJF\\dm$\"+IEk&H)F\\dm7$$\"+e%GpL$F\\dm$\"+]P4V\"*F\\dm7$$\"+:-V&\\$ F\\dm$\"++<7B**F\\dm7$$\"+ZhUkOF\\dm$\"+$p%yx5Facm7$$\"+ " 0 "" {MPLTEXT 1 0 56 "xx := evalf(Pi);\n evalf(g(xx),15);\nevalf(evalf(h(%),15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+aEfTJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0S,ivT /@)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "xx := evalf(-Pi);\nevalf(g(xx),15);\nevalf( evalf(h(%),15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!+aEfTJ!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0KBHmugz'!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$!+aEfTJ!\"*" }}}{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 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 272 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = dy/dx;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F% F(*$%\"xGF&F(!\"\"*&%#dyGF(%#dxGF," }{TEXT -1 4 " , " }{XPPEDIT 18 0 "y(1) = 2;" "6#/-%\"yG6#\"\"\"\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(1) = 5;" "6#/-%$y~'G6#\"\"\"\"\"&" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "de := x*diff(y(x),x$2)=diff(y(x),x);\nic := y(1)=2,D(y)(1)=5;\ndes olve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&%\"x G\"\"\"-%%diffG6$-%\"yG6#F'-%\"$G6$F'\"\"#F(-F*6$F,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"\"\"\"#/--%\"DG6#F(F)\"\"&" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(\"\"&\"\"\"\"\"#!\" \"F'F,F+#F+F,F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 38 "___________ ___________________________" }}{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 38 "______________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 5 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 5 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = x/``(dy/dx);" "6#/*(%\"dG\"\"#%\"yG\"\" \"*&F%F(*$%\"xGF&F(!\"\"*&F+F(-%!G6#*&%#dyGF(%#dxGF,F," }{TEXT -1 4 " \+ , " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`y '`(0) = 3;" "6#/-%$y~'G6#\"\"!\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 127 "de := diff(y(x),x$2)=x/diff(y(x),x);\nic := y(0)=0 ,D(y)(0)=3;\ndesolve(\{de,ic\},y(x));\ng := unapply(rhs(%),x);\nplot(g (x),x=-8..8);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 38 "_____________ _________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 38 "______________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 5 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 5 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2)+2*``(dy/dx)^2 = y^2;" "6#/,&*(%\"dG\"\"#% \"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&F'F)*$-%!G6#*&%#dyGF)%#dxGF-F'F)F) *$F(F'" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\" !F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 1/4;" "6#/-%$y~'G6# \"\"!*&\"\"\"F)\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "de := diff( y(x),x$2)+2*diff(y(x),x)^2=y(x)^2;\nic := y(0)=0,D(y)(0)=1/4;\ndesolve (\{de,ic\},y(x));\n#h := unapply(lhs(subs(y(x)=y,%)),y);\n#plots[impli citplot](x=h(y),x=-4..6,y=-5..4);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#deG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2)-F(6$F*F- F1F2F2*$)F*F1F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\" \"!F*/--%\"DG6#F(F)#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% $IntG6$,$*&\"\"%\"\"\",(*&\"\")F*)%#_uG\"\"#F*F**&F)F*F/F*!\"\"F*F*#F2 F0F*/F/;\"\"!-%\"yG6#%\"xGF:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 38 "_______________________________ _______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 38 "______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 273 1 "y" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)=1+``(dy/dx)^2" "6#/*(%\"dG\"\"# %\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\",&F(F(*$-%!G6#*&%#dyGF(%#dxGF,F&F(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 0;" "6#/-%$y~'G6#\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "de := y(x)*diff(y(x),x$2)=1+diff(y(x),x)^ 2;\nic := y(0)=1,D(y)(0)=0;\ndesolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/*&-%\"yG6#%\"xG\"\"\"-%%diffG6$F'-%\"$G6$F* \"\"#F+,&F+F+*$)-F-6$F'F*F2F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# icG6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&#\"\"\"\"\"#F+*&,&F+F+-%$expG6#,$*&F, F+F'F+F+F+F+-F06#,$F'!\"\"F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 38 "______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 38 "____________ __________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = y^2*``(dy/dx);" "6 #/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&F'F&-%!G6#*&%#dyGF(%# dxGF,F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\" !F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 1;" "6#/-%$y~'G6#\" \"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 334 "de := diff(y(x),x$2)=y(x)^2 *diff(y(x),x);\nic := y(0)=0,D(y)(0)=1;\ndesolve(\{de,ic\},y(x));\nh : = unapply(lhs(subs(y(x)=y,%)),y):\ngn := desolveK2(\{de,ic\},y(x),x=-2 ..1.74);\np1 := plots[implicitplot](x=h(y),x=-1.5..1.73,y=-1.4..8):\np 2 := plot('gn(x)',x=-1.5..1.73,color=green,thickness=2):\nplots[displa y]([p1,p2],view=[-1.5..1.73,-1.4..8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&)F)F0\"\"\"-F'6$F )F,F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--% \"DG6#F(F)\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&)\"\"$#\"\"\" F'F),&-%#lnG6#*$),&-%\"yG6#%\"xGF)*$F&F)F)F(F)F)-F,6#*&F)F)*$),(*$)F1 \"\"#F)F)*&F1F)F&F)!\"\"*$)F'#F>F'F)F)#F)\"\"'F)F@F)F)F)*&F(F)*&)F'#\" \"&FEF)-%'arctanG6#*&,&*&#F>\"\"*F)*&FBF)F1F)F)F)#F)F'F@F)F'#F)F>F)F)F )*(\"#=F@F'FI%#PiGF)F)F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 38 " ______________________________________" }}{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 38 "____________________ __________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }