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imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
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262 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 
1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 263 1 {CSTYLE "" 
-1 -1 "Helvetica" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 3 0 0 0 0 1 0 1 0 
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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 256 51 "First Order Differentia
l Equations\nAssorted Methods" }}{PARA 258 "" 0 "" {TEXT 257 14 "Brian
 E. Blank" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "
" {TEXT -1 83 "Click on a [+] sign to expand a section. Click on a [-]
 sign to collapse a section." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 "I
" }{TEXT 488 11 "ntroduction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 335 7 "In this" }{TEXT -1 2 "  " }{TEXT 258 5 "MAPLE" 
}{TEXT -1 4 "    " }{HYPERLNK 17 "worksheet" 2 "worksheet" "" }{TEXT 
334 14 ",  we will use" }{TEXT -1 2 "  " }{TEXT 259 5 "MAPLE" }{TEXT 
-1 3 "   " }{TEXT 333 75 "to implement several methods of solving firs
t order differential equations." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 259 "" 0 "" {TEXT 503 9 "Key Words" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 152 "The  w
ords and symbols that are used in this worksheet have been collected h
ere as hypertext links. Click on any word to go to the correspond help
 page." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 
17 "Dchangevar" 2 "Dchangevar" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "DEt
ools" 2 "DEtools" "" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "
dfieldplot" 2 "dfieldplot" "" }{TEXT -1 3 "   " }}{PARA 15 "" 0 "" 
{HYPERLNK 17 "diff" 2 "diff" "" }{TEXT 496 1 " " }}{PARA 15 "" 0 "" 
{HYPERLNK 17 "display" 2 "display" "" }{TEXT -1 1 " " }{TEXT 497 1 " \+
" }}{PARA 15 "" 0 "" {HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 2 "
  " }}{PARA 15 "" 0 "" {HYPERLNK 17 "for" 2 "for" "" }{TEXT -1 1 " " }
}{PARA 15 "" 0 "" {HYPERLNK 17 "implicitplot" 2 "implicitplot" "" }
{TEXT -1 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "int" 2 "int" "" }}
{PARA 15 "" 0 "" {HYPERLNK 17 "LambertW" 2 "LambertW" "" }}{PARA 15 "
" 0 "" {HYPERLNK 17 "loop" 2 "loop" "" }{TEXT -1 1 " " }}{PARA 15 "" 
0 "" {HYPERLNK 17 "lists" 2 "lists" "" }{TEXT -1 2 "  " }}{PARA 15 "" 
0 "" {HYPERLNK 17 "normal" 2 "normal" "" }{TEXT -1 3 "   " }}{PARA 15 
"" 0 "" {HYPERLNK 17 "odeadvisor" 2 "odeadvisor" "" }{TEXT -1 1 " " }}
{PARA 15 "" 0 "" {HYPERLNK 17 "odetest" 2 "odetest" "" }}{PARA 15 "" 
0 "" {HYPERLNK 17 "op" 2 "op" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "pack
ages" 2 "packages" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "plot" 2 "plot" 
"" }{TEXT 329 1 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "plots" 2 "plots" 
"" }}{PARA 15 "" 0 "" {HYPERLNK 17 "plot,structure" 2 "plot,structure
" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "restart" 2 "restart" "" }}{PARA 
15 "" 0 "" {HYPERLNK 17 "selection" 2 "selection" "" }}{PARA 15 "" 0 "
" {HYPERLNK 17 "solve" 2 "solve" "" }{TEXT -1 1 " " }}{PARA 15 "" 0 "
" {HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" 
{HYPERLNK 17 "value" 2 "value" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "wit
h\n" 2 "with" "" }}{PARA 0 "" 0 "" {TEXT 332 34 "On-line information a
bout using a " }{TEXT 328 6 " MAPLE" }{TEXT -1 2 "  " }{TEXT 330 59 "f
unction can also be obtained at a prompt by preceding the " }{TEXT 
327 5 "exact" }{TEXT 331 52 " keyword by a question mark and executing
 that line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "
" {TEXT 502 5 "Lists" }}{PARA 0 "" 0 "" {TEXT -1 89 "In this worksheet
 we will see several functions that return each several data items.  I
n " }{TEXT 504 5 "MAPLE" }{TEXT -1 5 " a   " }{HYPERLNK 17 "list" 2 "l
ist" "" }{TEXT -1 114 "  is an ordered sequence enclosed in square bra
ckets. For example, the list of square integers less than fifty is:" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "A := [seq(k^2, k = 1..7)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"AG7)\"\"\"\"\"%\"\"*\"#;\"#D\"#O\"#\\" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "If we want to select the j'th elem
ent we use the syntax   " }{TEXT 505 4 "A[j]" }{TEXT -1 1 ":" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A[6
];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 297 "Many of the functions that wil
l be given in this worksheet return a list of steps that are taken to \+
implement a particular differential equation solving procedure. Since \+
the last entry is the solution itself, it is convenient to have a func
tion that extracts that entry. Here is such a function:\n\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "lastEntry := proc()\n             \+
if nargs = 1 and type(args[1], list) then\n             RETURN( args[1
][nops(args[1])] );\n             else\n             RETURN(args);\n  \+
           fi:\n             end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "lastEntry(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#
\\" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 262 "" 0 "" {TEXT -1 
20 "Homgeneous Equations" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 358 6 "Theor
y" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 77 " S
uppose that we are to solve a first order differential equation of the
 form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 336 24 
"y '(x) = F( x , y(x) ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 342 19 "If  the expression " }{TEXT -1 1 " " }{TEXT 337 
15 " F( x , y(x) ) " }{TEXT -1 2 "  " }{TEXT 343 19 "can be expressed \+
as" }{TEXT -1 2 "  " }{TEXT 338 13 "Q( y(x)/x )  " }{TEXT -1 1 " " }
{TEXT 344 23 " then the substitution " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 339 13 "v(x) = y(x)/x" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 346 45 "leads to a first order s
eparable equation for" }{TEXT -1 2 "  " }{TEXT 341 1 "v" }{TEXT -1 1 "
 " }{TEXT 345 17 " as a function of" }{TEXT -1 2 "  " }{TEXT 340 1 "x
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 276 7 "Example" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 30 "Solve th
e homogeneous equation" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "
   " }}{PARA 0 "" 0 "" {TEXT -1 23 "                       " }
{XPPEDIT 278 0 "`y'(x)` = x*y/(x^2+y^2);" "/%&y'(x)G*(%\"xG\"\"\"%\"yG
F&,&*$F%\"\"#F&*$F'F*F&!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 279 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "restart:  with( plots ):  with( DEtools ):" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "diff_eq := d
iff(y(x),x) = x*y(x)/(x^2+y(x)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%(diff_eqG/-%%diffG6$-%\"yG6#%\"xGF,*(F,\"\"\"F)F.,&*$F,\"\"#F.*$F)F1
F.!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 347 
96 "Like many differential equations, this equation falls under severa
l types in the classification." }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "dsolve(diff_eq, y(x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&*&%\"xG\"\"#-%\"yG6#F&!\"##!\"\"F'-%#lnG6#F(\"\"\"%$
_C1G" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 348 3 "So " }{TEXT 
352 5 "MAPLE" }{TEXT -1 1 " " }{TEXT 353 56 " has found a solution - i
t involves a special function, " }{HYPERLNK 17 "LambertW" 2 "LambertW
" "" }{TEXT 349 20 ",  that is known to " }{TEXT 350 5 "MAPLE" }{TEXT 
-1 1 " " }{TEXT 351 31 ".  \nLet us verify the solution:" }{TEXT -1 1 
"\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "y(x) = solve(\" , y(x
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,&-%)Lam
bertWG6#*&F'\"\"#-F)6#,$%$_C1G!\"#\"\"\"#F6F0F4F6" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(\", dif
f_eq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%$expG6#,&-%)Lam
bertWG6#*&%\"xG\"\"#-F(6#,$%$_C1G!\"#\"\"\"#F6F0F4F6F/*(F/F6F'F6,&*$F/
F0F6*$F'F0F6!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simpli
fy(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,-%)LambertWG6#*&%\"xG\"
\"#-%$expG6#,$%$_C1G!\"#\"\"\"F1,&F1F1F%F1!\"\"F)F3-F,6#F/F1*(F)F*F+F1
F%F3#F1F*F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq(\");
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT 
354 27 "\nNow let us see how to use " }{TEXT 355 6 " MAPLE" }{TEXT 
356 54 "  to solve the the differential equation step-by-step:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
51 "diff_eq2 := Dchangevar(y(x)=x*v(x), [diff_eq], x );" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%)diff_eq2G/,&-%\"vG6#%\"xG\"\"\"*&F*F+-%%diffG
6$F'F*F+F+*(F*\"\"#F'F+,&*$F*F1F+*&F*F1F'F1F+!\"\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 57 "diff_eq3 := diff(v(x),x) = solve(diff_eq2, \+
diff(v(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)diff_eq3G/-%%diff
G6$-%\"vG6#%\"xGF,,$*(F)\"\"$F,!\"\",&\"\"\"F2*$F)\"\"#F2F0F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "diff_eq4 := lhs(diff_eq3) = \+
normal(rhs(diff_eq3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)diff_eq4G
/-%%diffG6$-%\"vG6#%\"xGF,,$*(F)\"\"$F,!\"\",&\"\"\"F2*$F)\"\"#F2F0F0
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "int((1+v^2)/v^3,v) = in
t(-1/x,x)+C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#%\"vG\"\"\"
*$F(!\"##!\"\"\"\"#,&-F&6#%\"xGF-%\"CGF)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "subs(v = y(x)/x, \");" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,&-%#lnG6#*&-%\"yG6#%\"xG\"\"\"F,!\"\"F-*&F)!\"#F,\"\"##F.F1,&-
F&F+F.%\"CGF-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 361 57 "Notice that this is the same solution as was obtained by
 " }{TEXT 365 8 " dsolve " }{TEXT 366 7 "(with  " }{TEXT 367 2 "C " }
{TEXT 368 12 "instead of  " }{TEXT 369 4 " _C1" }{TEXT 370 65 "). \n\n
After the next section we will give the code for a function," }{TEXT 
-1 1 " " }{TEXT 357 15 "homogeneous_ode" }{TEXT -1 2 ", " }{TEXT 362 
106 "that performs each of these steps in an automated fashion, and re
ports the results of the principal steps." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 363 8 "Graphics" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 364 103 "In this subsection w
e will plot some solution curves and superimpose them on the direction
 field plot. " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 116 "direction_field_plot := dfieldplot( [diff_eq] , [y(x
)], x = -5 .. 5 , y = 0 .. 4 ,color = COLOR(RGB,0.7,0.85,0.75)):" }}}
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45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 402 2 
0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" 
{TEXT 359 8 "Code for" }{TEXT -1 1 " " }{TEXT 280 15 "homogeneous_ode
" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 493 78 "You must execute th
is function here in order to use it later in the worksheet!" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4066 "
homogeneous_ode := proc()\noption `Copyright \2511997-2001 Brian E. Bl
ank,  All Rights Reserved.`:\nlocal xx, yy, x_0, y_0, expr, expr_uv, n
u, eqn, eqn2, eqn3, eqn4, eqn5, eqn6, Eqn5, u_, v_, cc, LIST1, LIST2, \+
LIST3, NF1, NF2, NF3, jj, kk, nu_integrand, xx_integrand, xi,eta;\nglo
bal _C;\nreadlib(isolate):\n\nif nargs < 1 or nargs > 2 then \nERROR(`
homogeneous_ode expects one or two arguments.`);\nfi:\n\nif nargs = 1 \+
and not type(args[1],equation) then \nERROR(`homogeneous_ode expects i
ts argument to be an equation.`):\nelif nargs = 2 and (not type(args[1
],equation) or not type(args[2],equation))\nthen\nERROR(`homogeneous_o
de expects its arguments to be equations.`):\nfi:\n\nif nargs = 2 then
\nif (not type( lhs(args[2]),function) or not type(op(0,lhs(args[2])),
 name) or nops( lhs(args[2]) ) <> 1 or not type(rhs(args[2]),algebraic
))\nthen ERROR(`homogeneous_ode expects its second argument to be of t
he form y(a) = c.`):\nelse \ny_0 := rhs(args[2]);\nx_0 := op(1,lhs(arg
s[2])):\nyy := op(0,lhs(args[2])):\nfi:\nfi:\n \n\nif not has(lhs(args
[1]), diff) then\nERROR(`homogeneous_ode expects its first argument to
 be of the form diff(y(x),x)=F(x,y(x)).`):\nelif nops(op(1,lhs(args[1]
))) <> 1\nthen\nERROR(`homogeneous_ode expects its first argument to b
e of the form diff(y(x),x)=F(x,y(x)) where y(x) is a function of one v
ariable.`):\nelif not type(op(1,(op(1,lhs(args[1])))),name)\nthen\nERR
OR(`homogeneous_ode expects its first argument to be of the form diff(
y(x),x)=F(x,y(x)) where y(x) is a function of one variable and x is a \+
name of a variable.`):\nelse\nxx := op(1,(op(1,lhs(args[1])))):\nexpr \+
:= rhs(args[1]):\nfi;\n\n\nif not type(rhs(args[1]), algebraic) then\n
ERROR(`homogeneous_ode expects its first argument to be of the form di
ff(y(x),x)=F(x,y(x)).`):\nfi:\n\nif nargs = 2 and op(0,op(1,lhs(args[1
])))<>yy\nthen\nERROR(`homogeneous_ode expects the dependent variable \+
names in the first two arguments to be the same.`):\nfi:\n\nif nargs =
 1 then yy := op(0,op(1,lhs(args[1]))): fi:\n\nexpr_uv := subs(\{xx = \+
u_, yy(xx)=v_\},expr):\n\nif simplify(expr_uv - subs(\{u_=t*u_,v_=t*v_
\},expr_uv)) <> 0 then \nERROR(`The differential equation does not app
ear to be homogeneous.`);\nfi:\n\n\neqn := args[1]:\neqn2 := DEtools[D
changevar](yy(xx)=xx*nu(xx) , [eqn], xx );\neqn2 := expand(lhs(eqn2))=
rhs(eqn2):\neqn3 := diff(nu(xx),xx) = solve(eqn2, diff(nu(xx),xx)):\ne
qn4 := lhs(eqn3) = normal(rhs(eqn3));\n\n\nif type(factor(numer(rhs(eq
n4))),`*`) then\n    LIST1 := [op(factor(numer(rhs(eqn4))))];\nelse\n \+
   LIST1 := [numer(rhs(eqn4))];\nfi;\nif type(denom(rhs(eqn4)),`*`) th
en\n   LIST2 := map(z->z^(-1),[op(factor(denom(rhs(eqn4))))]);\nelse\n
    LIST2 := [denom(rhs(eqn4))^(-1)];\nfi;\nNF1 := nops(LIST1);\nNF2 :
= nops(LIST2); \nLIST3 := [];\nfor kk from 1 to NF1 do\nif type(LIST1[
kk], `^`) and nops(op(1,LIST1[kk])) > 1 and type(op(1,LIST1[kk]),`*`) \+
then\n   for jj from 1 to nops(op(1,LIST1[kk])) do\n   LIST3 := [op(LI
ST3),op(jj,op(1,LIST1[kk]))^op(2,LIST1[kk])];\n   od;\nelse\nLIST3 := \+
[op(LIST3),LIST1[kk]];\nfi;\nod; \n\nfor kk from 1 to NF2 do\nif type(
LIST2[kk], `^`) and nops(op(1,LIST2[kk])) > 1 and type(op(1,LIST2[kk])
,`*`) then\n   for jj from 1 to nops(op(1,LIST2[kk])) do\n   LIST3 := \+
[op(LIST3),op(jj,op(1,LIST2[kk]))^(-op(2,LIST2[kk]))];\n   od:\nelse\n
LIST3 := [op(LIST3),LIST2[kk]];\nfi;\nod;\n\nNF3 := nops(LIST3);\n\nnu
_integrand := 1;\nxx_integrand := 1;\n\nfor jj from 1 to NF3 do\nif ha
s(LIST3[jj],nu(xx) ) \n   then nu_integrand := nu_integrand*LIST3[jj]^
(-1);\nelse\n   xx_integrand := xx_integrand*LIST3[jj];\nfi;\nod;\n\n
\n\nnu_integrand := subs(nu(xx)=nu,nu_integrand);\n\nif nargs = 1\nor \+
(nargs = 2 and x_0 = 0)\nthen\nEqn5 :=\nInt(nu_integrand,nu)=Int(xx_in
tegrand,xx)+_C;\neqn5 :=\nint(nu_integrand,nu)=int(xx_integrand,xx)+_C
;\neqn6 :=\nsubs(nu = yy(xx)/xx,eqn5);\nRETURN( [eqn,yy(xx)=xx*nu(xx),
eqn2,eqn3,eqn4,Eqn5,eqn5,eqn6]);\n\nelse\nEqn5 :=Int(subs(nu=eta,nu_in
tegrand),eta=y_0/x_0..nu(xx))=Int(subs(xx=xi,xx_integrand),xi=x_0..xx)
;\neqn5 := int(subs(nu=eta,nu_integrand),eta=y_0/x_0..nu(xx))=int(subs
(xx=xi,xx_integrand),xi=x_0..xx);\neqn6 := subs(nu(xx) = yy(xx)/xx,eqn
5);\nRETURN([eqn,yy(xx)=xx*nu(xx),eqn2,eqn3,eqn4,Eqn5,eqn5,eqn6]);\nfi
:\n\nend:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}{SECT 0 {PARA 4 "" 0 "
" {TEXT 360 14 "Description of" }{TEXT -1 2 "  " }{TEXT 281 15 "homoge
neous_ode" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 
15 "homogeneous_ode" }{TEXT 30 76 " - Exhibit steps for solving a firs
t order homogeneous differential equation" }}{PARA 4 "" 0 "usage" 
{TEXT -1 18 "Calling Sequences:" }}{PARA 0 "" 0 "" {TEXT -1 5 "     " 
}{TEXT 282 47 "homogeneous_ode(ode)\n  homogeneous_ode(ode, ic)" }}
{PARA 4 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 
5 "     " }{TEXT 23 58 "ode    - an equation of the form diff(y(x),x) \+
= F(x,y(x)) " }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{TEXT 23 65 "ic   \+
  - an equation of the form y(a) = c (the initial condition)" }}{PARA 
0 "" 0 "" {TEXT 23 1 "\n" }{TEXT 283 12 "Description:" }}{PARA 15 "" 
0 "" {TEXT -1 9 "The call " }{TEXT 284 42 " homogeneous_ode(diff(y(x),
x) = F(x,y(x)))" }{TEXT -1 22 "  returns a list with\n" }{TEXT 285 1 "
7" }{TEXT -1 29 " entries. The first entry is " }{TEXT 326 24 "diff(y(
x),x) = F(x,y(x))" }{TEXT -1 128 " and the last entry is the solution \+
of this differential equation (in implicit equation form). Further det
ails will be provided." }}{PARA 15 "" 0 "" {TEXT -1 67 "The required f
orm of the equation passed as the first argument of  " }{TEXT 297 15 "
homogeneous_ode" }{TEXT -1 80 "  is strictly enforced. In particular, \+
the left hand side of the first equation " }{TEXT 287 4 "must" }{TEXT 
-1 17 " be of the form  " }{TEXT 286 12 "diff(y(x),x)" }{TEXT -1 60 " \+
 and nothing else. For example, the differential equations " }{TEXT 
288 38 "\n           diff(y(x),x)+x/y(x)=y(x)/x" }{TEXT -1 2 ", " }
{TEXT 289 32 " \n           x*diff(y(x),x)=y(x)" }{TEXT -1 7 ", \nand \+
" }{TEXT 290 28 "\n           1 = diff(y(x),x)" }{TEXT -1 139 ",\nwhil
e perfectly valid as homogeneous differential equations, nevertheless \+
result in error messages when passed as the first argument of  " }
{TEXT 298 15 "homogeneous_ode" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" 
{TEXT -1 4 "If  " }{TEXT 291 1 "x" }{TEXT -1 6 "  and " }{TEXT 292 2 "
 y" }{TEXT -1 108 "  are respectively the independent and dependent va
riables of the differential equation, then the expression" }{TEXT 294 
2 "y " }{TEXT -1 73 "cannot appear on either the left or right side of
 the first argument of  " }{TEXT 299 15 "homogeneous_ode" }{TEXT -1 
20 ". It must appear as " }{TEXT 293 4 "y(x)" }{TEXT -1 8 ". Thus, " }
{TEXT 295 21 "diff(y(x),x) = x/y(x)" }{TEXT -1 32 "  is a valid first \+
argument but " }{TEXT 296 19 " diff(y(x),x) = x/y" }{TEXT -1 9 "  is n
ot." }}{PARA 15 "" 0 "" {TEXT -1 9 "The call " }{TEXT 300 42 " homogen
eous_ode(diff(y(x),x) = F(x,y(x)))" }{TEXT -1 97 "  results in a check
 for homogeneity. To be specific, an attempt is made to verify the ide
ntity  " }{TEXT 301 17 "F(t*u,t*v)=F(u,v)" }{TEXT -1 57 "  is examined
. If this identity cannot be verified, then " }{TEXT 302 41 "homogeneo
us_ode(diff(y(x),x) = F(x,y(x)))" }{TEXT -1 36 " returns a string with
 that message." }}{PARA 15 "" 0 "" {TEXT -1 40 "If the first order dif
ferential equation" }{TEXT 303 4 " ode" }{TEXT -1 18 "  is of the form
  " }{TEXT 309 24 "diff(y(x),x) = F(x,y(x))" }{TEXT -1 50 "  and if it
 passes the test for homogeneity, then " }{TEXT 304 20 "homogeneous_od
e(ode)" }{TEXT -1 46 " implements the change of variable\n\n          \+
" }{XPPEDIT 18 0 "y(x) = x*nu(x);" "/-%\"yG6#%\"xG*&F&\"\"\"-%#nuG6#F&
F(" }{TEXT -1 56 ". \n\nThis equation is the second entry of the retur
n of  " }{TEXT 306 20 "homogeneous_ode(ode)" }{TEXT -1 36 ". The third
 entry of the return of  " }{TEXT 307 20 "homogeneous_ode(ode)" }
{TEXT -1 179 "is the differential equation that results when the afore
mentioned change of variable is made (but prior to any further algebra
ic manipulation). The fourth entry of the return of  " }{TEXT 308 20 "
homogeneous_ode(ode)" }{TEXT -1 27 "is obtained by isolating   " }
{XPPEDIT 18 0 "diff(nu(x),x);" "-%%diffG6$-%#nuG6#%\"xGF(" }{TEXT -1 
74 "  (without further algebraic manipulation).  The fifth entry retur
ned by  " }{TEXT 310 21 "homogeneous_ode(ode) " }{TEXT -1 75 "is obtai
ned by simplifying the right side of the differential equation in  " }
{XPPEDIT 18 0 "nu(x);" "-%#nuG6#%\"xG" }{TEXT -1 34 "  so that it appe
ars clearly as   " }{XPPEDIT 18 0 "diff(nu(x),x) = G(x)*H(nu(x));" "/-
%%diffG6$-%#nuG6#%\"xGF)*&-%\"GG6#F)\"\"\"-%\"HG6#-F'6#F)F." }{TEXT 
-1 2 " ." }}{PARA 15 "" 0 "" {TEXT -1 35 "The final two entries return
ed by  " }{TEXT 311 15 "homogeneous_ode" }{TEXT -1 58 "  depend on whe
ther it is called with one argument or two." }}{PARA 15 "" 0 "" {TEXT 
-1 90 "Continuing with the notation used in a previous bullet item, th
e sixth entry returned by  " }{TEXT 312 20 "homogeneous_ode(ode)" }
{TEXT -1 9 "  is     " }{XPPEDIT 18 0 "Int(1/H(nu),nu) = Int(G(x),x)+_
C;" "/-%$IntG6$*&\"\"\"F'-%\"HG6#%#nuG!\"\"F+,&-F$6$-%\"GG6#%\"xGF3F'%
#_CGF'" }{TEXT -1 51 " .  This is also the sixth entry of the return o
f  " }{TEXT 313 27 "homogeneous_ode(ode,y(0)=c)" }{TEXT -1 9 ".   If  \+
 " }{XPPEDIT 18 0 "a <> 0;" "0%\"aG\"\"!" }{TEXT -1 8 ", then  " }
{TEXT 314 27 "homogeneous_ode(ode,y(a)=c)" }{TEXT -1 11 "  returns  " 
}{XPPEDIT 18 0 "Int(1/H(eta),eta = c/a .. nu(x)) = Int(G(xi),xi = a ..
 x);" "/-%$IntG6$*&\"\"\"F'-%\"HG6#%$etaG!\"\"/F+;*&%\"cGF'%\"aGF,-%#n
uG6#%\"xG-F$6$-%\"GG6#%#xiG/F;;F1F5" }{TEXT -1 23 "    as its sixth en
try." }}{PARA 15 "" 0 "" {TEXT -1 21 "The final entry of   " }{TEXT 
315 20 "homogeneous_ode(ode)" }{TEXT -1 4 " or " }{TEXT 317 27 "homoge
neous_ode(ode,y(0)=c)" }{TEXT -1 48 "  is an equation that provides th
e solution to  " }{TEXT 316 3 "ode" }{TEXT -1 57 "  in implicit form i
nvolving a constant of integration,  " }{TEXT 318 2 "_C" }{TEXT -1 8 "
.  If   " }{XPPEDIT 18 0 "a <> 0;" "0%\"aG\"\"!" }{TEXT -1 8 ", then  \+
" }{TEXT 319 27 "homogeneous_ode(ode,y(a)=c)" }{TEXT -1 40 " solves fo
r the constant of integration." }}{PARA 15 "" 0 "" {TEXT -1 129 "No at
tempt is made to verify the correctness of the solution. It is recomme
nded that the user test for correctness with the call " }{TEXT 320 36 
"odetest(homogeneous_ode(ode)[7],ode)" }{TEXT -1 34 ". The result of t
his call will be " }{TEXT 321 1 "0" }{TEXT -1 27 "  if the solution fo
und by " }{TEXT 322 15 "homogeneous_ode" }{TEXT -1 12 " is correct." }
}{PARA 15 "" 0 "" {TEXT -1 63 "Because nu is used as the change of var
iable, the arguments of " }{TEXT 323 15 "homogeneous_ode" }{TEXT -1 
22 " should be free of nu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT 305 8 "Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 324 14 "First example:" }{TEXT -1 1 "\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "diff_eq := diff(y(x),x) = x*
y(x)/(x^2 + y(x)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(diff_eqG/-%
%diffG6$-%\"yG6#%\"xGF,*(F,\"\"\"F)F.,&*$F,\"\"#F.*$F)F1F.!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "homogeneous_ode(diff_eq);" }
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