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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 38 "Unit 4 -- First-Or
der Linear Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 
0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.math.sc.edu/~m
eade/" "" }}{PARA 258 "" 0 "" {URLLINK 17 "Industrial Mathematics Inst
itute" 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 259 "" 0 "" 
{URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc.edu/" "
" }}{PARA 260 "" 0 "" {URLLINK 17 "University of South Carolina" 4 "ht
tp://www.sc.edu/" "" }}{PARA 261 "" 0 "" {TEXT -1 19 "Columbia, SC 292
08\n" }}{PARA 263 "" 0 "" {TEXT -1 7 "URL:   " }{URLLINK 17 "http://ww
w.math.sc.edu/~meade/" 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  b
y Douglas B. Meade" }}{PARA 266 "" 0 "" {TEXT -1 19 "All rights reserv
ed" }}{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 4" }}
{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "4.A" 1 "" "4.A" }{TEXT -1 38 " St
ructure of Solutiosn to Linear ODEs" }}{PARA 0 "" 0 "" {HYPERLNK 17 "4
.B" 1 "" "4.B" }{TEXT -1 48 " Integrating Factor for a First-Order Lin
ear ODE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 14 "Initialization" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "with( DEtools );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "wit
h( plots );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg
 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 
3 "" 0 "4.A" {TEXT -1 41 "4.A Structure of Solutions to Linear ODEs" }
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "The general first-order linear OD
E is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "lin_ode := diff( x(t), t ) + p(t) * x(t) = f(t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "Note that, in general, this is " }{TEXT 256 3 "not" }
{TEXT -1 17 " a separable ODE." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "odeadvisor( lin_ode, [separa
ble] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "p(t)" "6#-%\"pG6#%\"t
G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT 
-1 94 " are both constants, then the ODE is separable. In this case, t
he solution can be found as in " }{HYPERLNK 17 "Unit 3" 1 "unit03.mws
" "" }{TEXT -1 31 ". The solution that is found is" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "lin_ode_con
st := subs( p(t)=a, f(t)=b, lin_ode );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "odeadvisor( lin_ode_const, [separable] );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "infolevel[dsolve] := 3:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "lin_ode_const_soln := dsolve( lin_ode_const, x(t) );
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 0:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 91 "The structure of this solution is important. The term inv
olving the constant of integration" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "soln_h := x(t) = coeff(rhs(
lin_ode_const_soln),_C1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "is a solution of the homogen
ous ODE (i.e., " }{XPPEDIT 18 0 "b=0" "6#/%\"bG\"\"!" }{TEXT -1 1 ")" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 46 "odetest( soln_h, subs( b=0, lin_ode_const ) );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 17 "The constant term" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "soln_p := x(t) = subs( _C1=0, rhs(l
in_ode_const_soln) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "is a solution to the non-homogen
eous ODE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "odetest( soln_p, lin_ode_const );" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 361 "An
y solution that satisfies the full ODE is called a particular solution
. It is a general property of linear equations that the general soluti
on can be written as the sum of the general solution to the homogeneou
s equation and any solution to the non-homogeneous equation. This stru
cture will appear again when higher-order ODEs and systems of ODEs are
 studied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "4.B" {TEXT -1 51 "4.B In
tegrating Factor for a First-Order Linear ODE" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 173 "Knowledge of the structure of solutions to linear ODEs
 is important, but does not provide too much information about finding
 solutions to the general first-order linear ODE." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "lin_ode;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 93 "The general procedure for solving a first-order linear OD
E is to find an integrating factor, " }{XPPEDIT 18 0 "mu(t)" "6#-%#muG
6#%\"tG" }{TEXT -1 40 ", for the ODE. That is, find a function " }
{XPPEDIT 18 0 "mu(t)" "6#-%#muG6#%\"tG" }{TEXT -1 41 " such that when \+
the ODE is multiplied by " }{XPPEDIT 18 0 "mu(t)" "6#-%#muG6#%\"tG" }
{TEXT -1 94 " the left-hand side of the resulting equation can be writ
ten as the derivative of the product " }{XPPEDIT 18 0 "mu(t) * x(t)" "
6#*&-%#muG6#%\"tG\"\"\"-%\"xG6#F'F(" }{TEXT -1 64 ". The general formu
la for the integrating factor for this ODE is" }}{PARA 274 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "mu(t) = exp( int( p(t), t ) )" "6#/-%#m
uG6#%\"tG-%$expG6#-%$intG6$-%\"pG6#F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 83 "The advantage that this presents is that the differe
ntial equation now has the form" }}{PARA 272 "" 0 "" {XPPEDIT 18 0 "di
ff( X(t), t ) = F(t)" "6#/-%%diffG6$-%\"XG6#%\"tGF*-%\"FG6#F*" }{TEXT 
-1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "X(t) = \+
mu(t)*x(t)" "6#/-%\"XG6#%\"tG*&-%#muG6#F'\"\"\"-%\"xG6#F'F," }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "F(t) = mu(t)*f(t)" "6#/-%\"FG6#%\"tG*&-%#
muG6#F'\"\"\"-%\"fG6#F'F," }{TEXT -1 89 ". The explicit general soluti
on of this equation can be found by direct integration to be" }}{PARA 
273 "" 0 "" {XPPEDIT 18 0 "X(t)" "6#-%\"XG6#%\"tG" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "Int( diff(X(t), t ), t )" "6#-%$IntG6$-%%diffG6$-%\"XG6
#%\"tGF,F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int( F(t), t ) + C" "6#,
&-%$IntG6$-%\"FG6#%\"tGF*\"\"\"%\"CGF+" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 61 "To conclude, the solution to the original ODE is fou
nd using " }{XPPEDIT 18 0 "x(t) = X(t)/mu(t)" "6#/-%\"xG6#%\"tG*&-%\"X
G6#F'\"\"\"-%#muG6#F'!\"\"" }{TEXT -1 89 ". Instead of writing the gen
eral formula, implement this approach for a specific problem." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Consider \+
the ODE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "lin_ode1 := diff( x(t), t ) + x(t)/(t+1) = ln(t)/(t+1
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 17 "In this problem, " }{XPPEDIT 18 0 "p(t)=1/(t+1)" "
6#/-%\"pG6#%\"tG*&\"\"\"F),&F'F)F)F)!\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "f(t) = ln(t)/(t+1)" "6#/-%\"fG6#%\"tG*&-%#lnG6#F'\"\"\"
,&F'F,F,F,!\"\"" }{TEXT -1 33 ". Thus, the integrating factor is" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 45 "int_fact := mu(t) = exp( Int( 1/(t+1), t ) );" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 31 "int_fact1 := value( int_fact );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 4 "The " }{HYPERLNK 17 "DEtools" 2 "DEtools" "" }{TEXT -1 18 
" package contains " }{HYPERLNK 17 "intfactor" 2 "DEtools,intfactor" "
" }{TEXT -1 77 ", a procedure that will find an integrating factor for
 problems of this type." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "intfactor( lin_ode1 );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "ode2 := subs( int_fact1, mu(t)*lin_ode1 );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 148 "While this equation is rather complicated, the definitio
n of the integrating factor allows us to replace the left-hand side wi
th a single derivative" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ode3 := subs( int_fact1, Diff( mu(t
)*x(t), t ) ) = rhs(ode2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This differential equation
 can be solved by direct integration" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "int_ode3 := map(Int, ode3
, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 147 "The left-hand side is trivial to evaluate, Map
le does a fine job with the right-hand side. The result, after adding \+
the constant of integration, is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "q1 := subs( int_fact1, mu(t)
*x(t) ) = int( rhs(ode3), t ) + C;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The explicit \+
general solution to this first-order linear ODE is" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "expl_soln :
= op(solve( q1, \{x(t)\} ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "That this solution satis
fies the original differential equation is confirmed with" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "odet
est( expl_soln, lin_ode1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "To emphasize the structure
 of this solution, the homogeneous and particular solutions are" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 44 "soln_h := x(t) = coeff( rhs(expl_soln), C );" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 44 "soln_p := x(t) = subs( C=0, rhs(expl_soln))
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 15 "as confirmed by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "odetest( soln_h, lhs(lin_ode
1)=0 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "odetest( soln_p,
 lin_ode1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " }{HYPERLNK 17 "ODE Power
tool Table of Contents" 1 "unit00.mws" "" }{TEXT -1 1 "]" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }