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{SECT 0 {EXCHG {PARA 269 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 25 "Unit 8 -- Exact Eq
uations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {URLLINK 
17 "Prof. Douglas B. Meade" 4 "http://www.math.sc.edu/~meade/" "" }}
{PARA 258 "" 0 "" {URLLINK 17 "Industrial Mathematics Institute" 4 "ht
tp://www.math.sc.edu/~IMI/" "" }}{PARA 259 "" 0 "" {URLLINK 17 "Depart
ment of Mathematics" 4 "http://www.math.sc.edu/" "" }}{PARA 260 "" 0 "
" {URLLINK 17 "University of South Carolina" 4 "http://www.sc.edu/" "
" }}{PARA 261 "" 0 "" {TEXT -1 19 "Columbia, SC 29208\n" }}{PARA 263 "
" 0 "" {TEXT -1 7 "URL:   " }{URLLINK 17 "http://www.math.sc.edu/~mead
e/" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 264 "" 0 "" {TEXT 
-1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 262 "" 0 "" {TEXT -1 38 "Copyright \251  2001  by Douglas B. Mea
de" }}{PARA 266 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 268 
"" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 67 "---------------
----------------------------------------------------" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Outline of Unit 8" }}{EXCHG 
{PARA 14 "" 0 "" {HYPERLNK 17 "8.A" 1 "" "8.A" }{TEXT -1 13 " Basic Th
eory" }}{PARA 256 "" 0 "" {HYPERLNK 17 "8.A-1" 1 "" "8.A-1" }{TEXT -1 
10 " Example 1" }}{PARA 14 "" 0 "" {HYPERLNK 17 "8.B" 1 "" "8.B" }
{TEXT -1 20 " Integrating Factors" }}{PARA 256 "" 0 "" {HYPERLNK 17 "8
.B-1" 1 "" "8.B-1" }{TEXT -1 10 " Example 2" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Initiali
zation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "
with( linalg ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 0 {PARA 3 "" 0 "8.A" {TEXT -1 16 "8.A Basic Theory" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 65 "The general form for a first-order exact \+
differential equation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "gen_exact_ode := P(x,y(x)) + Q(x,y(
x))*diff(y(x),x) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 192 "The special property of exact e
quations is that the left-hand side of the ODE can be written as an ``
exact derivative''. In this case, there is a function F=F(x,y) such th
at level curves of F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "level_curve := F(x,y(x)) = C;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 18 "for some constant " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 
-1 80 ", gives an implicit solution to the differential equation.That \+
is, the function " }{XPPEDIT 18 0 "F=F(x,y)" "6#/%\"FG-F$6$%\"xG%\"yG
" }{TEXT -1 14 " satisfies the" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "condition_to_be_exact := co
llect( diff( lhs(level_curve), x ) - lhs( gen_exact_ode ) = 0, diff(y(
x),x) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 70 "Careful consideration of this equation sh
ows that it is satisfied when" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "condition1 := isolate( eval(
 condition_to_be_exact, diff(y(x),x)=0 ), F );" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 82 "condition2 := isolate( coeff( lhs(condition_to_be_exa
ct), diff(y(x),x) ) = 0, F );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "That is, when" }}{PARA 
271 "" 0 "" {XPPEDIT 18 0 "Diff(F,x) = P" "6#/-%%DiffG6$%\"FG%\"xG%\"P
G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(F,y)=Q" "6#/-%%DiffG6$%\"F
G%\"yG%\"QG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 260 "For arbitrary pairs of functions P and Q, this
 condition will not be satisfied. However, assuming the function F has
 continuous second derivatives and the functions P and Q have continuo
us first derivatives, the equivalence of the mixed second derivatives \+
of F" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "equiv_of_mixed_deriv := D[1](D[2](F)(x,y(x))) = D[2](
D[1](F)(x,y(x)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "combines with the previous argumen
t" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "q1 := D[2](condition1);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "q2 := D[1](condition2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "to provide ne
cessary and sufficient conditions for an exact equation:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "eval
( equiv_of_mixed_deriv, [ D[2](condition1), D[1](condition2) ] );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 8 "That is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 73 "nec_and_suff_condition := (P,Q) -> diff( \+
Q(x,y), x ) = diff( P(x,y), y ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 
"nec_and_suff_condition(P,Q);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{SECT 0 {PARA 4 "" 0 "8.A-1" {TEXT -1 15 "8.A-1 Example 1" }
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Consider the differential equatio
n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "ode := exp(y(x)) + (x*exp(y(x))+cos(y(x))) * diff(y(x
),x) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 77 "To check if this ODE is exact, identify t
he coefficients for the general form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "P := unapply( eval( lhs(o
de), [diff(y(x),x)=0, y(x)=y] ), (x,y) );" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "Q := unapply( eval( coeff(lhs(ode),diff(y(x),x)), y(x
)=y ), (x,y) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "and check the necessary and suffic
ient condition for exactness:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "nec_and_suff_condition(P,Q);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 71 "Since this is an exact equation, we construct the so
lution in the form " }{XPPEDIT 18 0 "F(x,y)=C" "6#/-%\"FG6$%\"xG%\"yG%
\"CG" }{TEXT -1 8 " where  " }{XPPEDIT 18 0 "Diff( F(x,y), x ) = P(x,y
)" "6#/-%%DiffG6$-%\"FG6$%\"xG%\"yGF*-%\"PG6$F*F+" }{TEXT -1 5 " and \+
" }{XPPEDIT 18 0 "Diff( F(x,y), y ) = Q(x,y)" "6#/-%%DiffG6$-%\"FG6$%
\"xG%\"yGF+-%\"QG6$F*F+" }{TEXT -1 38 ". The first of these is satisfi
ed when" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "q1 := F(x,y) = int( P(x,y), x ) + g(y);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 40 "To satisfy the second condition requires" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "q2 := diff(
 rhs(q1), y ) - Q(x,y) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Thus," }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "q3 := isola
te( map( int, q2, y ), g(y) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "and so" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "soln_F :
= eval( rhs(q1), q3 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "and the implicit solution of th
e differential equation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "soln := soln_F = C;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 88 "A quick pause to check that this solution does satisfy the diff
erential equation reveals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "odetest( eval(soln,y=y(x)), ode );
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 77 "The function F is also called a first integral of th
e ODE. The Maple command " }{HYPERLNK 17 "firint" 2 "DEtools,firint" "
" }{TEXT -1 11 ", from the " }{HYPERLNK 17 "DEtools" 2 "DEtools" "" }
{TEXT -1 106 " package, can be used to express the solution of an exac
t ODE in terms of its first integral. For example," }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "firint( ode
 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 96 "Note that essentially the same solution is returne
d with either of the following single commands" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "exactsol( o
de, y(x) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dsolve( ode,
 y(x), [exact] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The even simpler command" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "i
nfolevel[dsolve] := 3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dsolve( o
de, y(x) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] :=
 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 126 "returns an equivalent, but different, solution. T
his is because Maple is able to find several classifications for this \+
example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "odeadvisor( ode );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "and uses a di
fferent solution method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "8.B" 
{TEXT -1 23 "8.B Integrating Factors" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 178 "Most first-order ODEs are not exact. Some of these ODEs can be
 made exact by multiplying the ODE by an appropriate function so that \+
the new ODE is exact. The integrating factor, " }{XPPEDIT 18 0 "mu(x,y
)" "6#-%#muG6$%\"xG%\"yG" }{TEXT -1 132 ", must be chosen so that the \+
necessary and sufficient condition is satisfied by the coefficients of
 the ODE after multiplication by " }{XPPEDIT 18 0 "mu(x,y)" "6#-%#muG6
$%\"xG%\"yG" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "unassign('P','Q','mu');" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "nec_and_suff_condition( mu*P, mu*Q \+
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 163 "This is a partial differential equation, but ther
e are numerous cases where the determination of the integrating factor
 can be completed under the assumption that " }{XPPEDIT 18 0 "mu" "6#%
#muG" }{TEXT -1 25 " is a function of either " }{XPPEDIT 18 0 "x" "6#%
\"xG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 92 ", \+
but not both. A key to getting a handle on this condition is to note w
hen at least one of " }{XPPEDIT 18 0 "Diff(Q,x)" "6#-%%DiffG6$%\"QG%\"
xG" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "Diff(P,y)" "6#-%%DiffG6$%\"PG
%\"yG" }{TEXT -1 9 " is zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "8.
B-1" {TEXT -1 15 "8.B-1 Example 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
88 "Consider the following example (and ignore the fact that it is a f
irst-order linear ODE)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "ode2 := (x+y(x)) - x * diff( y(x), \+
x ) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "P := (x,y) -> x*y-2;" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "Q := (x,y) -> x^2-x*y;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 52 "ode2 := P(x,y(x)) + Q(x,y(x)) * diff( y(x), x \+
) = 0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "That this equation is n
ot exact can be seen from" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "nec_and_suff_condition( P,Q );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "odeadvisor( ode2, [exact] );" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "A
n integrating factor, if one exists, must satisfy the partial differen
tial equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 47 "mu_pde := nec_and_suff_condition( mu*P, mu*Q );
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 142 "With a little inspection, it can be seen that this \+
PDE simplifies to a linear ODE when the integrating factor is assumed \+
to be independent of " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 1 ":" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "mu_ode := eval( mu_pde, mu(x,y)=mu(x) );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 56 "mu_ode2 := simplify( isolate( mu_ode, diff(mu(
x),x) ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 70 "The general solution to this separable (a
nd linear) ODE is found to be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "mu_sol := dsolve( mu_ode2, m
u(x) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 78 "Since any (non-trivial) solution can be u
sed as an integrating factor, choose " }{TEXT 19 3 "_C1" }{TEXT -1 13 
"=1 and define" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 47 "mu := unapply( eval( rhs(mu_sol), _C1=1 ), x )
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 67 "To confirm that this function is an integrating fact
or for the ODE," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 49 "simplify( nec_and_suff_condition( mu*P, mu*Q )
 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "odeadvisor( mu(x)*ode2, [exact] );
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 18 "The Maple command " }{HYPERLNK 17 "mutest" 2 "DEtool
s,mutest" "" }{TEXT -1 9 ", in the " }{HYPERLNK 17 "DEtools" 2 "DEtool
s" "" }{TEXT -1 118 " package, provides an alternate method for checki
ng the validity of an integrating factor for a differential equation:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "mutest( mu(x), ode2 );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }
{HYPERLNK 17 "intfactor" 2 "DEtools,intfactor" "" }{TEXT -1 24 " comma
nd, also from the " }{HYPERLNK 17 "DEtools" 2 "DEtools" "" }{TEXT -1 
105 " package, can be used as a black box to find an integrating facto
r. An integrating factor for this ODE is" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "intfactor( ode2 );" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 27 "The solution to this ODE is" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dsolve( mu(x)*ode
2, y(x), [exact], implicit );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Note that without the op
tional argument " }{TEXT 19 8 "implicit" }{TEXT -1 36 " Maple returns \+
the pair of solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "dsolve( mu(x)*ode2, y(x), [exact] )
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 75 "The specific solution obtained depends upon the spec
ific initial condition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 51 "The same solutions are also obtained from either of
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "exactsol( ode2 );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "dsolve( ode2, y(x), [exact] );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "In ter
ms of first integrals, the original ODE is not exact so does not have \+
a first integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 15 "firint( ode2 );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "But, when mul
tiplied by the integrating factor," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "q := firint( mu(x)*ode2 );
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 54 "which simplifies to the implicit solution found abov
e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "simplify( q );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " }
{HYPERLNK 17 "ODE Powertool Table of Contents" 1 "unit00.mws" "" }
{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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33 1 1 }