{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 263 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 264 "Tim es" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 265 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Empha sis" -1 266 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {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 49 "A procedure for solving 1st order Bernoulli DE's " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanai mo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 20 "Version: 10.10.2007 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " }{TEXT 0 7 "desolve" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " } {TEXT 266 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 20 "Bernoulli equations " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 48 "A first ord er differential equation of the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+p(x)*y = q(x)*y^nu;" "6#/,&*&%#dyG\"\"\"%#dx G!\"\"F'*&-%\"pG6#%\"xGF'%\"yGF'F'*&-%\"qG6#F.F')F/%#nuGF'" }{TEXT -1 15 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "nu" "6#%#nuG" }{TEXT -1 35 " is a real number, is is called a \+ " }{TEXT 263 18 "Bernoulli equation" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 4 "For " }{XPPEDIT 18 0 "nu=0" "6#/%#nuG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "nu=1" "6#/%#nuG\"\"\"" }{TEXT -1 39 " (i) is a linear differential equation." }}{PARA 0 "" 0 "" {TEXT -1 4 "For " } {XPPEDIT 18 0 "y<>0" "6#0%\"yG\"\"!" }{TEXT -1 33 ", (i) can be writte n in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^( -nu);" "6#)%\"yG,$%#nuG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+p( x)*y^(1-nu) = q(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"pG6#%\"xGF') %\"yG,&F'F'%#nuGF)F'F'-%\"qG6#F." }{TEXT -1 15 " ------- (ii). " }} {PARA 0 "" 0 "" {TEXT -1 8 "Suppose " }{XPPEDIT 18 0 "nu<>0" "6#0%#nuG \"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "nu<>1" "6#0%#nuG\"\"\"" } {TEXT -1 9 " and let " }{XPPEDIT 18 0 "u=y^(1-nu)" "6#/%\"uG)%\"yG,&\" \"\"F(%#nuG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then \+ " }{XPPEDIT 18 0 "du/dx=(1-nu)*y^(-nu)" "6#/*&%#duG\"\"\"%#dxG!\"\"*&, &F&F&%#nuGF(F&)%\"yG,$F+F(F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" " 6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "y^( -nu)" "6#)%\"yG,$%#nuG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1 /(1-nu)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&,&F&F&%#nuGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx" "6#*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 19 ", and (ii) becomes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(1-nu)" "6#*&\"\"\"F$,&F$F$%#nuG!\"\"F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "du/dx + p(x)*u=q(x)" "6#/,&*&%#duG\"\"\"%#dxG!\"\"F'*&- %\"pG6#%\"xGF'%\"uGF'F'-%\"qG6#F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "du/dx + (1-nu)*p(x )=(1-nu)*q(x)" "6#/,&*&%#duG\"\"\"%#dxG!\"\"F'*&,&F'F'%#nuGF)F'-%\"pG6 #%\"xGF'F'*&,&F'F'F,F)F'-%\"qG6#F0F'" }{TEXT -1 17 " ------- (iii). \+ " }}{PARA 0 "" 0 "" {TEXT -1 28 "which is a linear equation. " }} {PARA 0 "" 0 "" {TEXT -1 37 "Solving (iii) and using the relation " } {XPPEDIT 18 0 "u=y^(1-nu)" "6#/%\"uG)%\"yG,&\"\"\"F(%#nuG!\"\"" } {TEXT -1 27 ", gives a solution of (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "A procedure for solving 1st order DE's of Berno ulli type: " }{TEXT 0 9 "desolveBN" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 16 "desolveBN: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 18 "Calling Sequence:\n" }} {PARA 0 "" 0 "" {TEXT 260 2 " " }{TEXT -1 76 " desolveBN( de ) \n \+ desolveBN( \{de,ic\} )\n desolveBN( \{de,ic\},y(x) )" }{TEXT 261 1 "\n" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters: " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " \+ de - " }{TEXT -1 92 " a first order differential equation with the derivative given in the form diff(y(x),x)," }}{PARA 0 "" 0 "" {TEXT -1 94 " (if x and y are the independent a nd dependent variables respectively)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 23 10 " ic - " }{TEXT -1 50 " an initial condition in the form y(x0) = y0. " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }} {PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 9 "des olveBN" }{TEXT -1 54 " attempts to solve a Bernoulli differential equa tion: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx+p(x) *y = q(x)*y^nu;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"pG6#%\"xGF'%\"yG F'F'*&-%\"qG6#F.F')F/%#nuGF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 28 "by making the substitution: " }{XPPEDIT 18 0 " _u = y^(1- nu)" "6#/%#_uG)%\"yG,&\"\"\"F(%#nuG!\"\"" }{TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 90 "If no initial condition is given, a general s olution with an arbitrary constant is sought." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 8 "Options:" }{TEXT -1 1 "\n " }}{PARA 0 "" 0 "" {TEXT -1 93 "info=true/false\nThe option info=true causes intermediate steps in the solution to be printed." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 263 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 155 "To \+ make the procedure active open the subsection, place the cursor anywhe re after the prompt [ > and press [Enter].\nYou can then close up the subsection. " }}{PARA 0 "" 0 "" {TEXT 263 4 "Note" }{TEXT -1 16 ": Th e procedure " }{TEXT 0 9 "desolveBN" }{TEXT -1 24 " requires the proce dure " }{TEXT 0 9 "desolveLN" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "It may be accessed using the general procedure " }{TEXT 0 7 "desolve" }{TEXT -1 18 " with the syntax: " }{TEXT 0 28 "desolve(. .,method=bernoulli)" }{TEXT -1 2 ". " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "desolveBN: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6675 "desolveBN := proc()\n lo cal ff,vars,derivs,x,yx,y,df,dff,drv,la,prntflg,Options,\n initco nd,de,ic,x0,y0,rsic,lsic,f0,u0,\n startopts,xx,yy,ee,soln,de1,ic1 ,t1,t2,\n px,qx,ux,n,m,sols,sol,i,yy0,gotpow,A,B,nvars;\n\n if \+ nargs>0 then \n ff := args[1]\n else\n error \"at least on e argument must be supplied\"\n end if;\n initcond := false;\n i f type(ff,\{set(equation),list(equation)\}) and nops(ff)=2 then\n \+ ff := map(_u -> if has(_u,D) then convert(_u,diff) else _u end if,ff) ;\n de := op(1,ff);\n ic := op(2,ff);\n if not has(de,d iff) then\n de := op(2,ff);\n ic := op(1,ff);\n e nd if;\n initcond := true;\n elif type(ff,equation) then\n \+ if has(de,'D') then de := convert(de,'diff') else de := ff end if;\n \+ else\n error \"the 1st argument, %1, is invalid .. it should be an equation or a set (or list) of 2 equations\",ff;\n end if;\n\n \+ startopts := 2;\n if nargs>1 then\n ee := args[2];\n if t ype(ee,function) and nops(ee)=1 then\n yy := op(0,ee);\n \+ xx := op(1,ee);\n if type(xx,name) and type(yy,name) then\n startopts := 3;\n else\n error \"the 2n d 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>=startopts then\n Options:=[args[startopts..nargs]];\n \+ if not type(Options,list(equation)) then\n error \"each o ptional argument must be an equation\"\n end if;\n if hasopt ion(Options,'info','prntflg','Options') then \n if prntflg<>tr ue 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,Opti ons),procname;\n end if;\n end if;\n\n derivs := indets(de,'s pecfunc(anything,diff)');\n if derivs=\{\} then\n error \"the 1 st argument, %1, is invalid .. it should be a differential equation or a set (or list) containing a differential equation and an initial con dition\",ff;\n end if;\n nvars := nops(indets(derivs,'name'));\n \+ if nvars<>1 then\n if nvars=0 then\n error \"there is a \+ problem with the independent variable occurring in the derivative(s)\" ;\n else\n error \"there should only be one independent v ariable in the differential equation\"\n end if;\n end if;\n \+ nvars := nops(indets(derivs,'anyfunc(name)'));\n if nvars<>1 then\n \+ if nvars=0 then\n error \"there is a problem with the dep endent variable occurring in the derivative(s)\"\n else\n \+ error \"there should only be one dependent variable in the differenti al equation\"\n end if;\n end if;\n\n if nops(derivs)<>1 then \n error \"there are too many derivatives in the differential equ ation .. note that the differential equation must be of order 1\"\n \+ end if;\n df := op(1,derivs);\n if type(df,function) and op(0,df)= diff and nops(df)=2 then\n yx := op(1,df);\n if not type(yx, anyfunc(name)) then\n error \"the 1st argument %1, in the deri vative, %2, is invalid .. it should be the 'unknown' dependent variabl e\",yx,df;\n end if; \n x := op(2,df);\n if not type(x, name) then\n error \"the 2nd argument %1, in the derivative, % 2, is invalid .. it should be the independent variable\",x,df;\n \+ end if; \n else\n error \"the derivative, %1, does not make sen se\",df;\n end if;\n\n y := op(0,yx);\n vars := indets(de,name); \n if member(y,vars) then\n error \"%1 and %2 cannot both appea r in the differential equation\",yx,y;\n end if;\n if op(1,yx)<>x \+ then\n error \"the derivative, %1, does not make sense\",df;\n \+ end if;\n\n if startopts=3 then \n if x<>xx or y<>yy then\n \+ error \"cannot solve the differential equation for %1\",ee;\n \+ end if;\n end if;\n \n if initcond then\n lsic := lhs(ic); \n if type(lsic,function) and op(0,lsic)=y and nops(lsic)=1 \n \+ and type(op(1,lsic),algebraic) then\n \+ x0 := op(1,lsic);\n if has(x0,\{x,y\}) then\n err or \"initial condition must not involve %1 or %2\",x,y;\n end \+ if;\n else\n error \"initial condition is not decipherabl e\"\n end if;\n rsic := rhs(ic);\n if type(rsic,algebra ic) then\n y0 := rsic;\n if has(y0,\{x,y\}) then\n \+ error \"initial condition must not involve %1 or %2\",x,y;\n \+ end if;\n else\n error \"initial condition is not \+ decipherable\"\n end if;\n end if;\n\n drv := solve(de,df);\n if nops([drv])<>1 then\n error \"cannot obtain a unique expres sion for the derivative\"\n end if;\n\n t1 := subs(yx=y,drv);\n \+ t1 := collect(t1,y);\n if patmatch(t1,A::algebraic*y+B::algebraic,'l a') then\n px := -subs(la,A);\n else\n error \"the DE is n ot of Bernoulli type\"\n end if;\n t2 := simplify(t1+px*y);\n\n \+ gotpow := false;\n if patmatch(t2,A::algebraic*y^B::algebraic,'la') \+ then\n n := subs(la,B);\n qx := subs(la,A);\n gotpow := true;\n end if;\n if not gotpow or member(y,indets(qx,'function(n ame)')) or \n indets(n,name) intersect \{x,y\}<>\{\} then\n er ror \"the DE is not of Bernoulli type\"\n end if;\n \n ux := _u( x);\n m := 1-n;\n de1 := diff(ux,x)+m*px*ux=m*qx;\n\n if prntflg then\n print(``);\n print(`Bernoulli DE . . `,diff(yx,x)+p x*yx=qx*yx^n);\n print(`or . . `,``(yx^(-n))*diff(yx,x)+px*yx^m=q x);\n print(``);\n print(`Substituting . . `,ux=yx^m,` and ` ,diff(ux,x)=``(m*yx^(-n))*diff(yx,x),` gives . . `);\n print(``); \n print(``(1/m)*diff(ux,x)+px*ux=qx); \n end if; \n \n if initcond then\n u0 := simplify(y0^m); \n ic1 := _u(x0)=u0; \n soln := desolveLN(\{de1,ic1\},ux,info=prntflg);\n else\n \+ soln := desolveLN(de1,ux,info=prntflg);\n end if;\n\n if prntflg then\n print(soln); \n print(``)\n end if;\n\n if m<>-2 then\n soln := yx=simplify(rhs(soln)^(simplify(rationalize(1/m)) ));\n else\n sols := [solve(y^(-2)=rhs(soln),y)];\n if ini tcond then\n # Check which of the 2 solutions fits the initial condition.\n sol := []; \n for i to 2 do\n \+ yy0 := traperror(simplify(eval(subs(x=x0,sols[i]))));\n \+ if yy0<>lasterror and signum(y0-yy0)=0 then\n sol := [o p(sol),yx=simplify(sols[i])];\n else\n yy0 := traperror(limit(sols[i],x=x0));\n if yy0<>lasterror and signum(y0-yy0)=0 then\n sol := [op(sol),yx=simplify( sols[i])];\n end if;\n end if;\n end \+ do;\n if sol<>[] then soln := op(sol)\n else soln := s implify(subs(y=yx,soln)) end if;\n else\n soln := op(sols );\n end if;\n end if; \n return soln; \nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Examples are given in the following section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 28 "desolve(..,method=bernoulli)" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx+y/x = x*y^2;" "6#/,&*&%#dyG\"\" \"%#dxG!\"\"F'*&%\"yGF'%\"xGF)F'*&F,F'*$F+\"\"#F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 4 " or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/y^2" "6#*&\"\"\"F$*$%\"yG\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx + 1/(x*y) = x" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*& F'F'*&%\"xGF'%\"yGF'F)F'F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The substitution " }{XPPEDIT 18 0 "u=1/y" "6#/%\"uG*&\"\"\"F&%\"yG!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "du/dx = -1/(y^2);" "6#/*&%#duG\"\"\"%#dxG!\"\",$*&F&F&* $%\"yG\"\"#F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\" \"%#dxG!\"\"" }{TEXT -1 8 ", gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "-du/dx+u/x = x;" "6#/,&*&%#duG\"\"\"%#dxG!\"\"F)*&% \"uGF'%\"xGF)F'F," }{TEXT -1 3 "., " }}{PARA 0 "" 0 "" {TEXT -1 3 "or \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx-u/x=-x" "6# /,&*&%#duG\"\"\"%#dxG!\"\"F'*&%\"uGF'%\"xGF)F),$F,F)" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 93 "This is a linear differential equatio n which can be solved by the integrating factor method. " }}{PARA 0 " " 0 "" {TEXT -1 26 "The integrating factor is " }{XPPEDIT 18 0 "exp(In t(-1/x,x)=exp(-ln(x))" "6#-%$expG6#/-%$IntG6$,$*&\"\"\"F,%\"xG!\"\"F.F --F$6#,$-%#lnG6#F-F." }{XPPEDIT 18 0 "`` = 1/x" "6#/%!G*&\"\"\"F&%\"xG !\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Multiplying both sides of the last equation by " }{XPPEDIT 18 0 "1/x" "6#*&\"\"\"F$%\" xG!\"\"" }{TEXT -1 7 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/x" "6#*&\"\"\"F$%\"xG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "du/dx-u/x^2 = -1" "6#/,&*&%#duG\"\"\"%#dxG!\"\"F'*&%\"u GF'*$%\"xG\"\"#F)F),$F'F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6 #*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ u/x ] = -1 " "6#/7#*&%\"uG\"\"\"%\"xG!\"\",$F'F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "u/x = -x + c" "6#/*&%\"uG\"\"\"%\"xG!\"\",&F'F(%\"cGF& " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u= -x^2+c*x" "6#/%\"uG,&*$%\"xG\"\"# !\"\"*&%\"cG\"\"\"F'F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "which gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y =1/u" "6#/%\"yG*&\"\"\"F&%\"uG!\"\"" }{XPPEDIT 18 0 "``=1/(-x^2+c*x)" "6#/%!G*&\"\"\"F&,&*$%\"xG\"\"#!\"\"*&%\"cGF&F)F&F&F+" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "de := diff(y(x),x)+y(x)/x=x* y(x)^2;\ndesolve(de,y(x),method=bernoulli,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*&F*F.F-! \"\"F.*&F-F.)F*\"\"#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%3Bernoulli~DE~.~.~~G/,&-%%diffG6$-%\" yG6#%\"xGF,\"\"\"*&F)F-F,!\"\"F-*&F,F-)F)\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%(or~.~.~G/,&*&-%!G6#*&\"\"\"F+*$)-%\"yG6#%\"xG\"\"#F+! \"\"F+-%%diffG6$F.F1F+F+*&F+F+*&F1F+F.F+F3F+F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~. ~.~G/-%#_uG6#%\"xG*&\"\"\"F*-%\"yGF'!\"\"%&~and~G/-%%diffG6$F%F(*&-%!G 6#,$*&F*F**$)F+\"\"#F*F-F-F*-F16$F+F(F*%,~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%!G6# !\"\"\"\"\"-%%diffG6$-%#_uG6#%\"xGF1F*F**&F1F)F.F*F*F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE ~.~.~~G/,&-%%DiffG6$-%#_uG6#%\"xGF,\"\"\"*&F,!\"\"F)F-F/,$F,F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$,$*&\"\"\"F,%\"xG!\"\"F. F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G*&\"\"\"F %%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&%\"xG!\"\"-%#_uG6#F%\"\"\"-%$IntG6$F&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&% \"xG!\"\"-%#_uG6#F%\"\"\",&F%F&&%\"CG6#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#_uG6#%\"xG,& *$)F'\"\"#\"\"\"!\"\"*&&%\"CG6#F,F,F'F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$ *&\"\"\"F**&F'F*,&F'F*&%\"CG6#F*!\"\"F*F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&\"\"\"F**&,&F'F*%$ _C1G!\"\"F*F'F*F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = y+y^3;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&%\"yGF&*$F*\"\"$F&" }{TEXT -1 13 ", y(1) = 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "de := diff(y(x),x)=y(x)+y(x)^3;\nic := y(1)=2 ;\ndesolve(\{de,ic\},y(x),method=bernoulli,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&F)\"\"\"*$)F)\" \"$F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3Bernoulli~DE~.~.~~G/,&-%%diffG6$-%\"yG6#%\"xGF,\"\"\"F)!\"\"*$ )F)\"\"$F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%(or~.~.~G/,&*&-%!G6#*& \"\"\"F+*$)-%\"yG6#%\"xG\"\"$F+!\"\"F+-%%diffG6$F.F1F+F+*&F+F+*$)F.\" \"#F+F3F3F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_uG6#%\"xG*&\"\"\"F**$)-%\"yG F'\"\"#F*!\"\"%&~and~G/-%%diffG6$F%F(*&-%!G6#,$*&F/F*F-!\"$F0F*-F46$F- F(F*%,~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%!G6##!\"\"\"\"#\"\"\"-%%diffG6$-%#_uG6#% \"xGF3F,F,F0F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$-%#_uG6#%\"xGF,\" \"\"*&\"\"#F-F)F-F-!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~factor~.~.~~G-%$expG6#-% $IntG6$\"\"#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~ ~=~~~G-%$expG6#,$*&\"\"#\"\"\"%\"xGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_uG6#%\"xG\"\"\"-%$ expG6#,$*&\"\"#F)F(F)F)F)-%$IntG6$,$*&F/F)F*F)!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_uG6# %\"xG\"\"\"-%$expG6#,$*&\"\"#F)F(F)F)F),&F*!\"\"&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApp lying~the~initial~condition~.~.~~G/&%\"CG6#\"\"\",$*&#\"\"&\"\"%F(-%$e xpG6#\"\"#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#_uG6#%\"xG,&\"\"\"!\"\"*&#\"\"&\"\"%F)*&-%$exp G6#\"\"#F)-F16#,$*&F3F)F'F)F*F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&\"\"#\"\"\", &\"\"%!\"\"*&\"\"&F+-%$expG6#,&F*F+*&F*F+F'F+F.F+F+#F.F*F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "de := diff(y(x),x)=y(x)+y(x)^3;\nic := y(1)=2;\ndsolve(\{de,ic\},y(x)): \nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-% \"yG6#%\"xGF,,&F)\"\"\"*$)F)\"\"$F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" yG6#%\"xG,$*&\"\"#\"\"\",&\"\"%!\"\"*&\"\"&F+-%$expG6#,&F*F+*&F*F+F'F+ F.F+F+#F.F*F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl e 3" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx+y = ex p(x)*y^2;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF'*&-%$expG6#%\"xGF'*$F *\"\"#F'" }{TEXT -1 13 ", y(0) = 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "de := diff(y(x),x)+y(x) =exp(x)*y(x)^2;\nic := y(0)=1;\ndesolve(\{de,ic\},y(x),method=bernoull i,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-% \"yG6#%\"xGF-\"\"\"F*F.*&-%$expGF,F.)F*\"\"#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3Bernoulli~DE~. ~.~~G/,&-%%diffG6$-%\"yG6#%\"xGF,\"\"\"F)F-*&-%$expGF+F-)F)\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%(or~.~.~G/,&*&-%!G6#*&\"\"\"F+*$)-%\" yG6#%\"xG\"\"#F+!\"\"F+-%%diffG6$F.F1F+F+*&F+F+F.F3F+-%$expGF0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 '%2Substituting~.~.~G/-%#_uG6#%\"xG*&\"\"\"F*-%\"yGF'!\"\"%&~and~G/-%% diffG6$F%F(*&-%!G6#,$*&F*F**$)F+\"\"#F*F-F-F*-F16$F+F(F*%,~gives~.~.~G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%!G6#!\"\"\"\"\"-%%diffG6$-%#_uG6#%\"xGF1F*F*F.F*-%$expGF0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$-%#_uG6#%\"xGF,\"\"\"F)!\"\",$-%$ expGF+F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$!\"\"%\"xG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G-%$expG6#,$ %\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_uG6#%\"xG\"\"\"-%$expG6#,$F(!\"\"F)-%$IntG6$F.F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_uG6#%\"xG\"\"\"-%$expG6#,$F(!\"\"F),&F(F.&%\"CG6#F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%EApplying~the~initial~condition~.~.~~G/&%\"CG6#\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#_uG6 #%\"xG,&*&-%$expGF&\"\"\"F'F,!\"\"F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&-%$expG 6#,$F'!\"\"\"\"\",&F'F/F/F.F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "de := diff(y(x),x)+y(x)=exp( x)*y(x)^2;\nic := y(0)=1;\ndsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"F*F.*&-%$expG F,F.)F*\"\"#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"! \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&-%$expG6#, $F'!\"\"\"\"\",&F'F/F/F.F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 4" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "dy/dx+tan(x)*y = tan(x)*y^3;" "6#/,&*&%#dyG\"\"\"%#dxG! \"\"F'*&-%$tanG6#%\"xGF'%\"yGF'F'*&-F,6#F.F'*$F/\"\"$F'" }{TEXT -1 13 ", y(0) = 3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "de := diff(y(x),x)+tan(x)*y(x)=tan(x)*y(x)^3;\n ic := y(0)=3;\ndesolve(\{de,ic\},y(x),method=bernoulli,info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\" \"*&-%$tanGF,F.F*F.F.*&F0F.)F*\"\"$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%3Bernoulli~DE~.~.~~G/,&-%%diffG6$-% \"yG6#%\"xGF,\"\"\"*&-%$tanGF+F-F)F-F-*&F/F-)F)\"\"$F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%(or~.~.~G/,&*&-%!G6#*&\"\"\"F+*$)-%\"yG6#%\"xG\" \"$F+!\"\"F+-%%diffG6$F.F1F+F+*&-%$tanGF0F+F.!\"#F+F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substitut ing~.~.~G/-%#_uG6#%\"xG*&\"\"\"F**$)-%\"yGF'\"\"#F*!\"\"%&~and~G/-%%di ffG6$F%F(*&-%!G6#,$*&F/F*F-!\"$F0F*-F46$F-F(F*%,~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*& -%!G6##!\"\"\"\"#\"\"\"-%%diffG6$-%#_uG6#%\"xGF3F,F,*&-%$tanGF2F,F0F,F ,F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$-%#_uG6#%\"xGF,\"\"\"*( \"\"#F--%$tanGF+F-F)F-!\"\",$*&F/F-F0F-F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~fa ctor~.~.~~G-%$expG6#-%$IntG6$,$*&\"\"#\"\"\"-%$tanG6#%\"xGF-!\"\"F1" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G*$)-%$cosG6#% \"xG\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_uG6#%\"xG\"\"\")-%$cosGF'\"\"#F)-%$IntG6$,$ *(F-F)F+F)-%$sinGF'F)!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_uG6#%\"xG\"\"\")-%$cosGF'\"\"# F),&*$F*F)F)&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~~ G/&%\"CG6#\"\"\"#!\")\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#_uG6#%\"xG,&\"\"\"F)*&#\"\")\"\"*F )*&F)F)*$)-%$cosGF&\"\"#F)!\"\"F)F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*(\"\"$\"\"\", &*&\"\"*F+)-%$cosGF&\"\"#F+F+\"\")!\"\"#F4F2F0F+F+" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "de := diff (y(x),x)+tan(x)*y(x)=tan(x)*y(x)^3;\nic := y(0)=3;\ndsolve(\{de,ic\},y (x)):\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%dif fG6$-%\"yG6#%\"xGF-\"\"\"*&-%$tanGF,F.F*F.F.*&F0F.)F*\"\"$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*(\"\"$\"\"\",&*&\"\"*F+)-%$cosGF&\" \"#F+F+\"\")!\"\"#F4F2F0F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "de := diff(y(x),x)+tan(x)*y (x)=tan(x)*y(x)^3;\ndesolve(de,y(x),method=bernoulli,info=false);\ndso lve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6 #%\"xGF-\"\"\"*&-%$tanGF,F.F*F.F.*&F0F.)F*\"\"$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&,&*$)-%$cosG6#%\"xG\"\"#\"\"\"F,&%\"CG6#F,F,#!\"\"F+F 'F,,$F#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG*&,&*$)-%$c osGF&\"\"#\"\"\"F/%$_C1GF/#!\"\"F.F,F//F$,$F(F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx+x*y(x) = x*y(x)^6;" "6#/,&*&%#dyG\"\"\"% #dxG!\"\"F'*&%\"xGF'-%\"yG6#F+F'F'*&F+F'*$-F-6#F+\"\"'F'" }{TEXT -1 12 ", y(0) = 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 99 "de := diff(y(x),x)+x*y(x)=x*y(x)^6;\nic := y(0 )=2;\ndesolve(\{de,ic\},y(x),method=bernoulli,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*&F-F.F *F.F.*&F-F.)F*\"\"'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG 6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%3Bernoulli~DE~.~.~~G/,&-%%diffG6$-%\"yG6#%\"xGF,\"\" \"*&F,F-F)F-F-*&F,F-)F)\"\"'F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%(or ~.~.~G/,&*&-%!G6#*&\"\"\"F+*$)-%\"yG6#%\"xG\"\"'F+!\"\"F+-%%diffG6$F.F 1F+F+*&F1F+F.!\"&F+F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_uG6#%\"xG*&\"\"\"F* *$)-%\"yGF'\"\"&F*!\"\"%&~and~G/-%%diffG6$F%F(*&-%!G6#,$*&F/F*F-!\"'F0 F*-F46$F-F(F*%,~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%!G6##!\"\"\"\"&\"\"\"-%%diffG6$ -%#_uG6#%\"xGF3F,F,*&F3F,F0F,F,F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$- %#_uG6#%\"xGF,\"\"\"*(\"\"&F-F,F-F)F-!\"\",$*&F/F-F,F-F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integra ting~factor~.~.~~G-%$expG6#-%$IntG6$,$*&\"\"&\"\"\"%\"xGF-!\"\"F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G-%$expG6#,$*(\" \"&\"\"\"\"\"#!\"\"%\"xGF+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_uG6#%\"xG\"\"\"-%$expG6#,$*(\" \"&F)\"\"#!\"\"F(F0F1F)-%$IntG6$,$*(F/F)F(F)F*F)F1F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_uG6#% \"xG\"\"\"-%$expG6#,$*(\"\"&F)\"\"#!\"\"F(F0F1F),&F*F)&%\"CG6#F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%EApplying~the~initial~condition~.~.~~G/&%\"CG6#\"\"\"#!#J\"#K" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%#_uG6#%\"xG,&\"\"\"F)*&#\"#J\"#KF)-%$expG6#,$*(\"\"&F)\"\"#!\"\"F' F4F)F)F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&\"\"#\"\"\",&\"#KF+*&\"#JF+-%$expG6#, $*(\"\"&F+F*!\"\"F'F*F+F+F6#F6F5F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "de := diff(y(x),x)+x*y(x)=x *y(x)^6;\nic := y(0)=2;\ndsolve(\{de,ic\},y(x)):\nsimplify(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\" \"*&F-F.F*F.F.*&F-F.)F*\"\"'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i cG/-%\"yG6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"x G,$*&\"\"#\"\"\",&\"#KF+*&\"#JF+-%$expG6#,$*(\"\"&F+F*!\"\"F'F*F+F+F6# F6F5F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx+x*y(x) = x* y(x)^(1/2);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"xGF'-%\"yG6#F+F'F'*&F +F')-F-6#F+*&F'F'\"\"#F)F'" }{TEXT -1 12 ", y(0) = 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "de := \+ diff(y(x),x)+x*y(x)=x*sqrt(y(x));\nic := y(0)=2;\ndesolve(\{de,ic\},y( x),method=bernoulli,info=true);\nexpand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*&F-F.F*F.F.*&F -F.F*#F.\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"! \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3Bernoulli~DE~.~.~~G/,&-%%diffG6$-%\"yG6#%\"xGF,\"\"\" *&F,F-F)F-F-*&F,F-F)#F-\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%(or~. ~.~G/,&*&-%!G6#*&\"\"\"F+*$-%\"yG6#%\"xG#F+\"\"#!\"\"F+-%%diffG6$F-F0F +F+*&F0F+F-#F+F2F+F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_uG6#%\"xG*$-%\"yGF' #\"\"\"\"\"#%&~and~G/-%%diffG6$F%F(*&-%!G6#,$*&F,F-*&F-F-*$F*#F-F.!\" \"F-F-F--F26$F*F(F-%,~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%!G6#\"\"#\"\"\"-%%diffG6$ -%#_uG6#%\"xGF1F*F**&F1F*F.F*F*F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$- %#_uG6#%\"xGF,\"\"\"*&#F-\"\"#F-*&F,F-F)F-F-F-,$*&F0!\"\"F,F-F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$,$*&\"\"#!\"\"%\"xG\"\" \"F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G-%$ex pG6#,$*&\"\"%!\"\"%\"xG\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_uG6#%\"xG\"\"\"-%$expG6# ,$*&\"\"%!\"\"F(\"\"#F)F)-%$IntG6$,$*&#F)F1F)*&F(F)F*F)F)F)F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*&-%#_uG6#%\"xG\"\"\"-%$expG6#,$*&\"\"%!\"\"F(\"\"#F)F),&F*F)&%\"CG6 #F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~~G/&%\"CG6#\"\"\", &*$\"\"##F(F+F(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#_uG6#%\"xG,&\"\"\"F)*&,&*$\"\"##F) F-F)F)!\"\"F)-%$expG6#,$*&\"\"%F/F'F-F/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$ ),(\"\"\"F+*&-%$expG6#,$*&\"\"%!\"\"F'\"\"#F3F+F4#F+F4F+F-F3F4F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,\"\"\"F)*(\"\"#F)-%$ex pG6#,$*&\"\"%!\"\"F'F+F2F)F+#F)F+F)*&F+F)F,F)F2*&\"\"$F))F,F+F)F)*(F+F )F7F)F+F3F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "de := diff(y(x),x)+x*y(x)=x*y(x)^(1/2);\nic := y(0 )=2;\ndsolve(\{de,ic\},y(x));\nallvalues(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*&F-F.F*F.F.*&F -F.F*#F.\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"! \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'RootOfG6#,** $%#_ZG#\"\"\"\"\"#!\"\"F/F/*&-%$expG6#,$*&\"\"%F1F'F0F1F/F0F.F/F3F1" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,\"\"\"F)*(\"\"#F)-%$e xpG6#,$*&\"\"%!\"\"F'F+F2F)F+#F)F+F)*&F+F)F,F)F2*&\"\"$F))F,F+F)F)*(F+ F)F7F)F+F3F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl e 7" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx+y(x) = y(x)^(sqrt(3)-1);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'-%\"yG6#%\"xGF')-F+ 6#F-,&-%%sqrtG6#\"\"$F'F'F)" }{TEXT -1 12 ", y(0) = 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "de := \+ diff(y(x),x)+y(x)=y(x)^(sqrt(3)-1);\nic := y(0)=2;\ndesolve(\{de,ic\}, y(x),method=bernoulli,info=true);\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"F*F. )F*,&*$-%%sqrtG6#\"\"$F.F.F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#icG/-%\"yG6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%3Bernoulli~DE~.~.~~G/,&-%%diffG6$-%\" yG6#%\"xGF,\"\"\"F)F-)F),&*$-%%sqrtG6#\"\"$F-F-F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%(or~.~.~G/,&*&-%!G6#)-%\"yG6#%\"xG,&\"\"\"F0*$-%% sqrtG6#\"\"$F0!\"\"F0-%%diffG6$F+F.F0F0)F+,&\"\"#F0F1F6F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Sub stituting~.~.~G/-%#_uG6#%\"xG)-%\"yGF',&\"\"#\"\"\"*$-%%sqrtG6#\"\"$F. !\"\"%&~and~G/-%%diffG6$F%F(*&-%!G6#*&F,F.)F*,&F.F.F/F4F.F.-F86$F*F(F. %,~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%!G6#*&\"\"\"F*,&\"\"#F**$-%%sqrtG6#\"\"$F*! \"\"F2F*-%%diffG6$-%#_uG6#%\"xGF9F*F*F6F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~ ~G/,&-%%DiffG6$-%#_uG6#%\"xGF,\"\"\"*&,&\"\"#F-*$-%%sqrtG6#\"\"$F-!\" \"F-F)F-F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$,&\"\"# \"\"\"*$-%%sqrtG6#\"\"$F,!\"\"%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%5~~~~~~~~~~~~~~~~=~~~G-%$expG6#,$*&,&*$-%%sqrtG6#\"\"$\"\"\"F/\"\"#! \"\"F/%\"xGF/F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_uG6#%\"xG\"\"\"-%$expG6#,$*&,&*$-%%sqrtG6# \"\"$F)F)\"\"#!\"\"F)F(F)F6F)-%$IntG6$,$*&F/F)F*F)F6F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#_uG6# %\"xG\"\"\"-%$expG6#,$*&,&*$-%%sqrtG6#\"\"$F)F)\"\"#!\"\"F)F(F)F6F),&F *F)&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~~G/&%\"CG6#\" \"\",&)\"\"#,$*$-%%sqrtG6#\"\"$F(!\"\"\"\"%F(F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#_uG6#%\"xG,& \"\"\"F)*&-%$expG6#*&,&*$-%%sqrtG6#\"\"$F)F)\"\"#!\"\"F)F'F)F),&)F5,$F 0F6\"\"%F)F6F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG),(\"\"\"F**(\"\"%F*-%$expG6#*&,&* $-%%sqrtG6#\"\"$F*F*\"\"#!\"\"F*F'F*F*)F7,$F2F8F*F*F-F8,&F7F*F2F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(),(\"\"\"F.*(\"\"%F.-%$expG6#*&,&*$-%%sqrtG6#\"\"$F.F.\"\"#!\"\"F.9 $F.F.)F;,$F6F " 0 "" {MPLTEXT 1 0 117 "de := diff(y(x),x)+y(x)=y (x)^(sqrt(3)-1);\nic := y(0)=2;\ngn := desolve(\{de,ic\},y(x),x=-3..5, type=numeric,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/, &-%%diffG6$-%\"yG6#%\"xGF-\"\"\"F*F.)F*,&*$-%%sqrtG6#\"\"$F.F.F.!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The two s olutions can be compared graphically . . . 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{TEXT -1 5 " 2 " }{XPPEDIT 18 0 "dy/dx = y/x-x/(y^2) ;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%\"yGF&%\"xGF(F&*&F,F&*$F+\"\"#F(F( " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 1" "6#/-%\"yG6#\"\"\"F'" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________ ______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 4 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 267 1 " x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+4*y = x^4*y^2;" "6#/,&*&%#dyG 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" " }}{PARA 0 "" 0 "" {TEXT -1 37 "_________ ____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }{XPPEDIT 18 0 "dy/dx+5*y = 3*sqrt(y);" "6#/,&*&%#dyG\"\"\"%#dxG!\" \"F'*&\"\"&F'%\"yGF'F'*&\"\"$F'-%%sqrtG6#F,F'" }{TEXT -1 4 ", " } {XPPEDIT 18 0 "y(0) = 1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 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