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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 72 "First Order Differential \+
Equations (Classification and Solving Practice)" }}{PARA 19 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 3 "1. " }{XPPEDIT 
19 1 "diff(y(x),x) = 2*x/(9*y^2);" "6#/-%%diffG6$-%\"yG6#%\"xGF**(\"\"
#\"\"\"F*F-*&\"\"*F-*$F(F,F-!\"\"" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 3 "2. " }{XPPEDIT 19 1 "diff(y(x),x) = x*sqrt(1-y)/sqrt(1-x^2);" "6
#/-%%diffG6$-%\"yG6#%\"xGF**(F*\"\"\"-%%sqrtG6#,&F,F,F(!\"\"F,-F.6#,&F
,F,*$F*\"\"#F1F1" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 3 "3. " }
{XPPEDIT 19 1 "(sqrt(x^2-y^2)+y)*dx-x*dy = 0;" "6#/,&*&,&-%%sqrtG6#,&*
$%\"xG\"\"#\"\"\"*$%\"yGF-!\"\"F.F0F.F.%#dxGF.F.*&F,F.%#dyGF.F1\"\"!" 
}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 3 "4. " }{XPPEDIT 19 1 "cos(y)*dx-
(x*sin(y)-y^2)*dy = 0;" "6#/,&*&-%$cosG6#%\"yG\"\"\"%#dxGF*F**&,&*&%\"
xGF*-%$sinG6#F)F*F**$F)\"\"#!\"\"F*%#dyGF*F5\"\"!" }}}{SECT 1 {PARA 3 
"" 0 "" {TEXT -1 3 "5. " }{XPPEDIT 19 1 "diff(y(x),x) = (sin(x)-3*x^2*
y)/(x^3);" "6#/-%%diffG6$-%\"yG6#%\"xGF**&,&-%$sinG6#F*\"\"\"*(\"\"$F0
*$F*\"\"#F0F(F0!\"\"F0*$F*F2F5" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 3 
"6. " }{XPPEDIT 19 1 "diff(y(x),x) = x*(1-y^4)/(y^3);" "6#/-%%diffG6$-
%\"yG6#%\"xGF**(F*\"\"\",&F,F,*$F(\"\"%!\"\"F,*$F(\"\"$F0" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 3 "7. " }{XPPEDIT 19 1 "x*diff(y(x),x) = x*ex
p(y/x)+y;" "6#/*&%\"xG\"\"\"-%%diffG6$-%\"yG6#F%F%F&,&*&F%F&-%$expG6#*
&F+F&F%!\"\"F&F&F+F&" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 3 "8. " }
{XPPEDIT 19 1 "tan(theta)*diff(r(theta),theta)-r = tan(theta)^2;" "6#/
,&*&-%$tanG6#%&thetaG\"\"\"-%%diffG6$-%\"rG6#F)F)F*F*F/!\"\"*$-F'6#F)
\"\"#" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 3 "9. " }{XPPEDIT 19 1 "x*d
iff(y(x),x)+y = x^3;" "6#/,&*&%\"xG\"\"\"-%%diffG6$-%\"yG6#F&F&F'F'F,F
'*$F&\"\"$" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 4 "10. " }{XPPEDIT 19 
1 "diff(y(x),x) = 3*exp(-2*x)-2*y;" "6#/-%%diffG6$-%\"yG6#%\"xGF*,&*&
\"\"$\"\"\"-%$expG6#,$*&\"\"#F.F*F.!\"\"F.F.*&F4F.F(F.F5" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 15 "Classifications" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(DEtools):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "1.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 30 "diff(y(x),x) = 2*x/(9*y(x)^2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%%diffG6$-%\"yG6#%\"xGF*,$*&F*\"\"\"*$)F'\"\"#F-!\"\"#F0\"\"*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "odeadvisor(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%+_separableG" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Separable equations are a
lso exact. We can write\n\n" }{TEXT 256 17 "                 " }
{XPPEDIT 257 1 "diff(y(x),x) = g(x)*h(y(x));" "6#/-%%diffG6$-%\"yG6#%
\"xGF**&-%\"gG6#F*\"\"\"-%\"hG6#-F(6#F*F/" }{TEXT -1 1 "\n" }}{PARA 0 
"" 0 "" {TEXT -1 2 "as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 21 "                     " }{XPPEDIT 258 1 "g(x)*dx-1*dy
/h(y) = 0;" "6#/,&*&-%\"gG6#%\"xG\"\"\"%#dxGF*F**(F*F*%#dyGF*-%\"hG6#%
\"yG!\"\"F2\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" 
{TEXT -1 49 "This equation satisfies the test for exactness:\n\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "testeq( diff(g(x),y) = diff(
-1/h(y), x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "because each term
 is 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
2 "2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "diff(y(x),x) = x*sqrt(1-
y(x))/sqrt(1-x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"y
G6#%\"xGF**&*&F*\"\"\"-%%sqrtG6#,&F-F-F'!\"\"F-F-*$-F/6#,&*$)F*\"\"#F-
F2F-F-F-F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "odeadvisor(%)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%+_separableG" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "3." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "(sqrt(x^2-y^2)+y)*dx-x*dy = 0;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,&*&,&*$-%%sqrtG6#,&*$)%\"xG\"\"#\"\"\"F0*$
)%\"yGF/F0!\"\"F0F0F3F0F0%#dxGF0F0*&F.F0%#dyGF0F4\"\"!" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 10 "Not exact:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "diff(sqrt(x^2-y^2)+y,y) , di
ff(-x,x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*&%\"yG\"\"\"*$-%%sqrt
G6#,&*$)%\"xG\"\"#F&F&*$)F%F/F&!\"\"F&F2F2F&F&F2" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "These two expressions are
 clearly different: the equation is not exact." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Let us rewrite it:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "di
ff(y(x),x) = (sqrt(x^2-y(x)^2)+y(x))/x;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%%diffG6$-%\"yG6#%\"xGF**&,&*$-%%sqrtG6#,&*$)F*\"\"#\"\"\"F5*$
)F'F4F5!\"\"F5F5F'F5F5F*F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "odeadvisor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$%-_homogeneo
usG%(class~AG7$%+_1st_orderG%8_with_linear_symmetriesG%*_rationalG%+_d
AlembertG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 192 "Let us ignore the fancy classification categories. This equati
on is homogeneous. Let us see how to verify this assertion for ourselv
es.  We write the right side as a function of two variables:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "subs(y(x)=y, (sqrt(x^2-y(x)^2)+y(x)
)/x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$-%%sqrtG6#,&*$)%\"xG\"
\"#\"\"\"F.*$)%\"yGF-F.!\"\"F.F.F1F.F.F,F2" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "The key for homogeneity is to \+
see if  k   survives when  x and  y   are replaced by  k*x  and  k*y  \+
respectively:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(k>0);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(\{x=k*x,y=k*y\}, %
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$-%%sqrtG6#*&)%#k|irG\"\"#
\"\"\",&*&F*F-)%\"xGF,F-F-*&F*F-)%\"yGF,F-!\"\"F-F-F-*&F*F-F4F-F-F-*&F
*F-F1F-F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$-%%sqrtG6#,&*$)%\"xG\"\"#\"\"
\"F.*$)%\"yGF-F.!\"\"F.F.F1F.F.F,F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "4
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "cos(y)*dx-(x*sin(y)-y^2)*dy \+
= 0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 6 "E
xact:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(cos(y),y);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$sinG6#%\"yG!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "diff(-(x*sin(y)-y^2),x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$-%$sinG6#%\"yG!\"\"" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "They are the same so the \+
equation is exact." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "5." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 41 "diff(y(x),x) = (sin(x)-3*x^2*y(x))/(x^3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xGF**&,&-%$sinGF
)\"\"\"*(\"\"$F/)F*\"\"#F/F'F/!\"\"F/*$)F*F1F/F4" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "odeadvisor(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7#%(_linearG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 29 "We may rewrite it in the form" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "diff(y(x),x) + 3/x*y(x) =sin(x)/x^3; " }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/,&-%%diffG6$-%\"yG6#%\"xGF+\"\"\"*&*&\"\"$F,F(F,F
,F+!\"\"F,*&-%$sinGF*F,*$)F+F/F,F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 5 "Here:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "P := x -> 3/x;   Q := x -> sin(x)/x^3;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"PGR6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F.9$!
\"\"\"\"$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QGR6#%\"xG6\"6$
%)operatorG%&arrowGF(*&-%$sinG6#9$\"\"\"*$)F0\"\"$F1!\"\"F(F(F(" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "6." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 37 "diff(y(x),x) = x*(1-y(x)^4)/(y(x)^3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xGF**&*&F*\"\"\",&F-F-*$)F'\"\"%F
-!\"\"F-F-*$)F'\"\"$F-F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"odeadvisor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%+_separableG" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 60 "This equation is also Bernoulli, as is se
en by rewriting it:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "diff(y(x),
x)+ x*y(x) = x*y(x)^(-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%dif
fG6$-%\"yG6#%\"xGF+\"\"\"*&F+F,F(F,F,*&F+F,*$)F(\"\"$F,!\"\"" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "7." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "x*diff(y(x),x) = x*exp(y(x)/
x)+y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"xG\"\"\"-%%diffG6$-%
\"yG6#F%F%F&,&*&F%F&-%$expG6#*&F*F&F%!\"\"F&F&F*F&" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "odeadvisor(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%7$%-_homogeneousG%(class~AG7$%+_1st_orderG%8_with_lin
ear_symmetriesG%+_dAlembertG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 
"Rewrite this to verify that it is homogeneous." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "8." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "tan(theta)*d
iff(r(theta),theta)-r(theta) = tan(theta)^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&*&-%$tanG6#%&thetaG\"\"\"-%%diffG6$-%\"rGF(F)F*F*F.!
\"\"*$)F&\"\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "odeadvi
sor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%(_linearG" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The linear form is
:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "diff(r(theta),theta)+(-1/tan
(theta))*r(theta) = tan(theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&
-%%diffG6$-%\"rG6#%&thetaGF+\"\"\"*&F(F,-%$tanGF*!\"\"F0F." }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "9." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "x*
diff(y(x),x)+y(x) = x^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"xG
\"\"\"-%%diffG6$-%\"yG6#F&F&F'F'F+F'*$)F&\"\"$F'" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "odeadvisor(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7#%(_linearG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 19 "The linear form is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "diff(y(x),x)+(1/x)*y(x) = x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/,&-%%diffG6$-%\"yG6#%\"xGF+\"\"\"*&F(F,F+!\"\"F,*$)F+\"\"#F," }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "10." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " diff(y(x),x) = 3*exp(-
2*x)-2*y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"
xGF*,&-%$expG6#,$F*!\"#\"\"$*&\"\"#\"\"\"F'F4!\"\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "odeadvisor(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7#7$%(_linearG%(class~AG" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The linear form is" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 37 " diff(y(x),x) + 2*y(x) = 3*exp(-2*x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%diffG6$-%\"yG6#%\"xGF+\"\"\"*&\"
\"#F,F(F,F,,$-%$expG6#,$F+!\"#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 26 "Soluti
ons to the Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 2 "1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xGF*,$*&F*\"\"\"
*$)F'\"\"#F-!\"\"#F0\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 18 "Separable method:\n" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$)%\"yG\"\"#
\"\"\"F)-F%6$,$%\"xG#F*\"\"*F/" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 
20 "6#/,$*$)%\"yG\"\"$\"\"\"#F)F(,&*$)%\"xG\"\"#F)#F)\"\"*%\"CGF)" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "2." }}{EXCHG 
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xGF**&*&F*\"\"\"
-%%sqrtG6#,&F-F-F'!\"\"F-F-*$-F/6#,&*$)F*\"\"#F-F2F-F-F-F2" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Separable metho
d:\n" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$-
%%sqrtG6#,&F(F(%\"yG!\"\"F(F/F.-F%6$*&%\"xGF(*$-F+6#,&F(F(*$)F3\"\"#F(
F/F(F/F3" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*$-%%sqrtG6#,&
\"\"\"F*%\"yG!\"\"F*!\"#,&*$-F'6#,&F*F**$)%\"xG\"\"#F*F,F*F,%\"CGF*" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "3." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 
20 "6#/,&*&,&*$-%%sqrtG6#,&*$)%\"xG\"\"#\"\"\"F0*$)%\"yGF/F0!\"\"F0F0F
3F0F0%#dxGF0F0*&F.F0%#dyGF0F4\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Let us r
ewrite it:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xGF**&,&*$-%%sqrtG6#,&*$)F*\"\"#
\"\"\"F5*$)F'F4F5!\"\"F5F5F'F5F5F*F8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 59 "This equation is homogeneous.  Make t
he change of variable:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&F'\"\"\"-%\"vGF&F)" }}
}{PARA 0 "" 0 "" {TEXT -1 57 "\nDifferentiate with respect to x, using
 the product rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"
yG6#%\"xGF*,&-%\"vGF)\"\"\"*&F*F.-F%6$F,F*F.F." }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Substitute the right hand
 side of the original equation for the left side of this equation:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6
#/*&,&*$-%%sqrtG6#,&*$)%\"xG\"\"#\"\"\"F/*$)-%\"yG6#F-F.F/!\"\"F/F/F2F
/F/F-F5,&-%\"vGF4F/*&F-F/-%%diffG6$F7F-F/F/" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Substitute\n" }}{EXCHG {PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&F'\"\"\"-%\"vGF&F)" }}}
{PARA 0 "" 0 "" {TEXT -1 37 "\ninto the last differential equation." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "
6#/*&,&*$-%%sqrtG6#,&*$)%\"xG\"\"#\"\"\"F/*&F,F/)-%\"vG6#F-F.F/!\"\"F/
F/*&F-F/F2F/F/F/F-F5,&F2F/*&F-F/-%%diffG6$F2F-F/F/" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Simplify the left side:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$-%%sqrtG6#,&\"\"\"F**$)-%
\"vG6#%\"xG\"\"#F*!\"\"F*F*F-F*,&F-F**&F0F*-%%diffG6$F-F0F*F*" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6
#/*&%\"xG\"\"\"-%%diffG6$-%\"vG6#F%F%F&*$-%%sqrtG6#,&F&F&*$)F*\"\"#F&!
\"\"F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
19 "Separate variables:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$-%%sqrtG6#,&F(F(
*$)%\"vG\"\"#F(!\"\"F(F2F0-F%6$*&F(F(%\"xGF2F6" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Evaluate integrals:\n" }}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arcsinG6#%\"vG,&-%#lnG6#%\"
xG\"\"\"%\"CGF-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "Substitute back for y:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arcsinG6#*&-%\"yG6#%\"xG\"
\"\"F+!\"\",&-%#lnGF*F,%\"CGF," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "4
." }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%$cosG6#%\"yG\"\"\"%
#dxGF*F**&,&*&%\"xGF*-%$sinGF(F*F**$)F)\"\"#F*!\"\"F*%#dyGF*F5\"\"!" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Since \+
this equation is exact, there is a function of two variables that sati
sfies the pair of equations:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 53 "diff(f(x,y),x)=cos(y),diff(f(x,y),y)=-(x*sin(y)-y^2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$/-%%diffG6$-%\"fG6$%\"xG%\"yGF*-%$cosG6#F+/-
F%6$F'F+,&*&F*\"\"\"-%$sinGF.F4!\"\"*$)F+\"\"#F4F4" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "We integrate the first
 of these equations with respect to x. (We do not forget about the sec
ond of these equations. We will need it soon.) Notice that when we und
o a partial derivative with respect to x, the \"constant\" of integrat
ion is a function of y:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%%diffG6$-%\"fG6$%\"xG%\"yG
F-F-,&-F%6$-%$cosG6#F.F-\"\"\"-%\"gGF4F5" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG,&*&-%$c
osG6#F(\"\"\"F'F.F.-%\"gGF-F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 81 "To determine the function g(y), we take t
he partial derivative with respect to y:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"fG6$%\"xG
%\"yGF+,&*&F*\"\"\"-%$sinG6#F+F.!\"\"-F%6$-%\"gGF1F+F." }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We turn to the diffe
rential equation that we have kept on reserve:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"
fG6$%\"xG%\"yGF+,&*&F*\"\"\"-%$sinG6#F+F.!\"\"*$)F+\"\"#F.F." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Equating \+
the right side of the last two equations, we obtain:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 40 " -x*sin(y)+y^2 = -x*sin(y)+diff(g(y),y);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"xG\"\"\"-%$sinG6#%\"yGF'!\"\"*
$)F+\"\"#F'F',&F%F,-%%diffG6$-%\"gGF*F+F'" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " diff(g(y),y)=y^2;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"gG6#%\"yGF**$)F*\"\"#
\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
22 "We solve this equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"yG,&*$)F'\"\"$\"\"\"#F,F+%
\"CGF," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
41 "and use it in our expression for  f(x,y):" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 28 "f(x,y) = cos(y)*x+1/3*y^3+C;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG,(*&-%$cosG6#F(\"\"\"F'F.F.*&#F.\"\"
$F.)F(F1F.F.%\"CGF." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 94 "Since the total differential of  f(x,y)  is 0, we have \+
 f(x,y) = A  (where A is some constant)" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&-%$cosG6#%\"yG\"\"\"%
\"xGF*F**&#F*\"\"$F*)F)F.F*F*%\"CGF*%\"AG" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "or, combining the two constants
 to obtain a third:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%$cosG6
#%\"yG\"\"\"%\"xGF*F**&#F*\"\"$F*)F)F.F*F*%\"KG" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 
"" {TEXT -1 2 "5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xGF**&,&-%$sinGF)\"\"\"*(
\"\"$F/)F*\"\"#F/F'F/!\"\"F/*$)F*F1F/F4" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Rewritten in linear form" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%
%diffG6$-%\"yG6#%\"xGF+\"\"\"*&*&\"\"$F,F(F,F,F+!\"\"F,*&-%$sinGF*F,*$
)F+F/F,F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 5 "Here:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"PG6#%\"xG,$*&\"\"\"F*F'!\"\"\"\"$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%\"QG6#%\"xG*&-%$sinGF&\"\"\"*$)F'\"\"$F+!\"\"
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "We m
ultiply each side of this linear equation by the integrating factor   \+
" }{XPPEDIT 19 1 "e^Int(P(x),x) = e^(3*ln(x));" "6#/)%\"eG-%$IntG6$-%
\"PG6#%\"xGF,)F%*&\"\"$\"\"\"-%#lnG6#F,F0" }{TEXT -1 1 ":" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "(Notice that  " }
{XPPEDIT 19 1 "e^(3*ln(x));" "6#)%\"eG*&\"\"$\"\"\"-%#lnG6#%\"xGF'" }
{TEXT -1 15 " simplifies to " }{XPPEDIT 19 1 "x^3;" "6#*$%\"xG\"\"$" }
{TEXT -1 3 " ):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"xG\"\"$
\"\"\",&-%%diffG6$-%\"yG6#F&F&F(*&*&F'F(F-F(F(F&!\"\"F(F(-%$sinGF/" }}
}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&)%\"xG\"\"$\"\"\"-%%diffG
6$-%\"yG6#F'F'F)F)*(F(F))F'\"\"#F)F-F)F)-%$sinGF/" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 202 "If you use this method, \+
then you must be able to use the Product Rule of Differentiation backw
ards to recognize the left side as the derivative of a product. That i
s the point of the integrating factor:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*&)%\"xG\"\"$
\"\"\"-%\"yG6#F*F,F*-%$sinGF/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 10 "Therefore:" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"xG\"\"$\"\"\"-%\"yG
6#F&F(-%$IntG6$-%$sinGF+F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"x
G\"\"$\"\"\"-%\"yG6#F&F(,&-%$cosGF+!\"\"%\"CGF(" }}}{PARA 0 "" 0 "" 
{TEXT -1 9 "\nFinally;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&,&-%$cosGF&\"\"\"%\"
CG!\"\"F-*$)F'\"\"$F-F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 2 "6." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%%diffG6$-%\"yG6#%\"xGF**&*&F*\"\"\",&F-F-*$)F'\"\"%F-!\"\"F-F-*$)F
'\"\"$F-F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "This equation is also Bernoulli
 with  " }{TEXT 261 6 "n = -3" }{TEXT -1 32 ", as is seen by rewriting
 it as:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%diffG6$-%\"yG6#%\"xG
F+\"\"\"*&F+F,F(F,F,*&F+F,*$)F(\"\"$F,!\"\"" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 
"We make the change of variable " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"vG6#%\"xG)-%\"yGF&,&\"\"\"
F,%\"nG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 7 "that is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"vG6#%\"xG*$)-%\"yGF&\"\"%\"\"\"" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "different
iate," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%%diffG6$-%\"vG6#%\"xGF*,$*&)-%\"yGF)\"\"$\"\"\"-F%6$
F.F*F1\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 37 "substitute the original equation for " }{XPPEDIT 19 1 "diff(y(x
),x);" "6#-%%diffG6$-%\"yG6#%\"xGF)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"vG6#%\"xGF*,$*&
)-%\"yGF)\"\"$\"\"\",&*&F*F1*$F-F1!\"\"F1*&F*F1F.F1F5F1\"\"%" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 12 "and simplify" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%%diffG6$-%\"vG6#%\"xGF*,$*&F*\"\"\",&!\"\"F-*$)-%\"yGF)\"\"%F-F-F
-!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 
"The point of this substitution was to achieve a linear equation in v(
x). " }}{PARA 0 "" 0 "" {TEXT -1 44 "We have one as can be seen by sub
stituting  " }{XPPEDIT 19 1 "y(x)^4 = v(x);" "6#/*$)-%\"yG6#%\"xG\"\"%
\"\"\"-%\"vG6#F)" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%%diffG6$-%\"vG6#%\"xGF*,$*&F*\"\"\",&!\"\"F-F'F-F-!\"%" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "and rewriting:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 
20 "6#/,&-%%diffG6$-%\"vG6#%\"xGF+\"\"\"*(\"\"%F,F+F,F(F,F,,$F+F." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The integ
rating factor is " }{XPPEDIT 19 1 "exp(Int(4*x,x));" "6#-%$expG6#-%$In
tG6$*&\"\"%\"\"\"%\"xGF+F," }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#-%$IntG6$
,$%\"xG\"\"%F,\"\"\",&-%%diffG6$-%\"vG6#F,F,F.*(F-F.F,F.F3F.F.F.,$*&F%
F.F,F.F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&-%$expG6#,$*$)%\"xG\"\"#\"\"\"F,F-,&-%%diffG6$-%\"vG
6#F+F+F-*(\"\"%F-F+F-F2F-F-F-,$*&F%F-F+F-F6" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&*&)-%$expG6#*$)%\"xG\"\"#\"\"\"F-F.-%%diffG6$-%\"vG6
#F,F,F.F.**\"\"%F.F&F.F,F.F2F.F.,$*&F&F.F,F.F6" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "We must identify the left
 side as a derivative of a product:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*&)-%$expG6#*$)%
\"xG\"\"#\"\"\"F0F1-%\"vG6#F/F1F/,$*&F)F1F/F1\"\"%" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "and therefore" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)-%$expG6#*$)%\"xG\"\"#\"\"\"F,F--%
\"vG6#F+F-,&-%$IntG6$,$*&F%F-F+F-\"\"%F+F-%\"CGF-" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "
" {XPPMATH 20 "6#/*&)-%$expG6#*$)%\"xG\"\"#\"\"\"F,F--%\"vG6#F+F-,&*$F
%F-F-%\"CGF-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"vG6#%\"xG,&\"\"\"F)*&%\"CGF)*$)-%$expG6#*$)F'
\"\"#F)F3F)!\"\"F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 8 "Finally," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$),&\"\"\"F+*&%\"CGF+*$
)-%$expG6#*$)F'\"\"#F+F5F+!\"\"F+#F+\"\"%F+" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "7." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"xG\"\"\"-%%diff
G6$-%\"yG6#F%F%F&,&*&F%F&-%$expG6#*&F*F&F%!\"\"F&F&F*F&" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Rewrite this to ex
hibit it as a homogeneous equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xGF*,&-%$
expG6#*&F'\"\"\"F*!\"\"F0F/F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Make the \+
change of variable:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&F'\"\"\"-%\"vGF&F)" }}}
{PARA 0 "" 0 "" {TEXT -1 57 "\nDifferentiate with respect to x, using \+
the product rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"
yG6#%\"xGF*,&-%\"vGF)\"\"\"*&F*F.-F%6$F,F*F.F." }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Substitute the right hand
 side of the original equation for the left side of this equation:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6
#/,&-%$expG6#*&-%\"yG6#%\"xG\"\"\"F,!\"\"F-F(F-,&-%\"vGF+F-*&F,F--%%di
ffG6$F0F,F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Substitute\n" }}{EXCHG 
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&F'\"\"\"-%\"vGF&F)" }}
}{PARA 0 "" 0 "" {TEXT -1 37 "\ninto the last differential equation." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 
"6#/,&-%$expG6#-%\"vG6#%\"xG\"\"\"F(F,,&F(F,*&F+F,-%%diffG6$F(F+F,F," 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "and si
mplify:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"vG6#%\"xGF*
*&-%$expG6#F'\"\"\"F*!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 105 "The point of the change of variable has been t
o achieve a separable equation. That has been accomplished." }}{PARA 
0 "" 0 "" {TEXT -1 19 "Separate variables:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(-
%$expG6#%\"vG!\"\"F,-F%6$*&F(F(%\"xGF-F1" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Evaluate integrals:\n" }}{EXCHG 
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"\"F&-%$expG6#%\"vG!\"\"F+,&-%
#lnG6#%\"xGF&%\"CGF&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 22 "Substitute back for y:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 
20 "6#/,$*&\"\"\"F&-%$expG6#*&-%\"yG6#%\"xGF&F.!\"\"F/F/,&-%#lnGF-F&%
\"CGF&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
37 "In this case we can solve explicitly:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&-%#lnG6#
,$*&\"\"\"F.,&-F*F&F.%\"CGF.!\"\"F2F.F'F." }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 2 "8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 
1 "" {XPPMATH 20 "6#/,&*&-%$tanG6#%&thetaG\"\"\"-%%diffG6$-%\"rGF(F)F*
F*F.!\"\"*$)F&\"\"#F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 36 "The linear form of this equation is:" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "
" 1 "" {XPPMATH 20 "6#/,&-%%diffG6$-%\"rG6#%&thetaGF+\"\"\"*&F(F,-%$ta
nGF*!\"\"F0F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The integrating factor is
    " }{XPPEDIT 19 1 "exp(Int(-1/tan(theta),theta));" "6#-%$expG6#-%$I
ntG6$,$*&\"\"\"F+-%$tanG6#%&thetaG!\"\"F0F/" }{TEXT -1 9 "   or    " }
{XPPEDIT 19 1 "exp(Int(-cot(theta),theta));" "6#-%$expG6#-%$IntG6$,$-%
$cotG6#%&thetaG!\"\"F-" }}{PARA 0 "" 0 "" {TEXT -1 4 "or  " }{XPPEDIT 
19 1 "exp(ln(csc(theta)));" "6#-%$expG6#-%#lnG6#-%$cscG6#%&thetaG" }}
{PARA 0 "" 0 "" {TEXT -1 5 "or   " }{XPPEDIT 19 1 "csc(theta);" "6#-%$
cscG6#%&thetaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 48 "Multiply the equation by the integrating \+
factor:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&*&-%$cscG6#%&thetaG\"\"\"-%%diffG6$-%\"rGF(F)F*F**&*
&F&F*F.F*F*-%$tanGF(!\"\"F4*&F&F*F2F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Ident
ify the left side as the derivative of a product:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$7#*&
-%\"rG6#%&thetaG\"\"\"-%$cscGF+F-F,*&F.F--%$tanGF+F-" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"rG6#%&th
etaG\"\"\"-%$cscGF'F),&-%$IntG6$*&F*F)-%$tanGF'F)F(F)%\"CGF)" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6
#/*&-%\"rG6#%&thetaG\"\"\"-%$cscGF'F),&-%#lnG6#,&-%$secGF'F)-%$tanGF'F
)F)%\"CGF)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"rG6#%&thetaG*&-%$si
nGF&\"\"\",&-%#lnG6#,&-%$secGF&F+-%$tanGF&F+F+%\"CGF+F+" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "9." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"
xG\"\"\"-%%diffG6$-%\"yG6#F&F&F'F'F+F'*$)F&\"\"$F'" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The linear form is:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%diffG6$-%\"yG6#%\"xGF+\"
\"\"*&F(F,F+!\"\"F,*$)F+\"\"#F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The integ
rating factor is " }{XPPEDIT 19 1 "exp(Int(1/x,x));" "6#-%$expG6#-%$In
tG6$*&\"\"\"F*%\"xG!\"\"F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 13 "or simply    " }{XPPEDIT 19 1 "x;" "6#%\"xG" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Multiply \+
the equation by this integrating factor:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%\"xG\"\"\",&-%%diffG6$-%\"yG6#F%F%F&*&F+F&F%!\"\"F&
F&*$)F%\"\"$F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 
1 "" {XPPMATH 20 "6#/,&*&%\"xG\"\"\"-%%diffG6$-%\"yG6#F&F&F'F'F+F'*$)F
&\"\"$F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
60 "We must identify the left side as a derivative of a product:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%%DiffG6$7#*&%\"xG\"\"\"-%\"yG6#F)F*F)*$)F)\"\"$F*" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "and therefore" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6
#/*&%\"xG\"\"\"-%\"yG6#F%F&,&-%$IntG6$*$)F%\"\"$F&F%F&%\"CGF&" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6
#/*&%\"xG\"\"\"-%\"yG6#F%F&,&*$)F%\"\"%F&#F&F-%\"CGF&" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&,&*$)F'\"\"%\"\"\"F.*&F-F.%\"CGF
.F.F.F'!\"\"#F.F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "10." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diff
G6$-%\"yG6#%\"xGF*,&-%$expG6#,$F*!\"#\"\"$*&\"\"#\"\"\"F'F4!\"\"" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The linea
r form is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%diffG6$-%\"yG6#%\"
xGF+\"\"\"*&\"\"#F,F(F,F,,$-%$expG6#,$F+!\"#\"\"$" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The integrating factor is
 " }{XPPEDIT 19 1 "exp(Int(2,x));" "6#-%$expG6#-%$IntG6$\"\"#%\"xG" }
{TEXT -1 9 "  or     " }{XPPEDIT 19 1 "e^(2*x);" "6#)%\"eG*&\"\"#\"\"
\"%\"xGF'" }{TEXT -1 41 ":   multiply the equation by this factor:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"eG,$%\"xG\"\"#\"\"\",&-%
%diffG6$-%\"yG6#F(F(F**&F)F*F/F*F*F*,$*&F%F*-%$expG6#,$F(!\"#F*\"\"$" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The le
ft side is the derivative of a product:" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%%DiffG6$7#*&-%$expG6#,$%\"xG\"\"#\"\"\"-%\"yG6#F-F/F-\"\"$" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$%\"xG\"\"#\"\"\"-
%\"yG6#F)F+,&-%$IntG6$\"\"$F)F+%\"CGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$%\"xG\"\"#\"\"\"
-%\"yG6#F)F+,&F)\"\"$%\"CGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&F'\"\"$%\"CG
\"\"\"F,-%$expG6#,$F'\"\"#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "12 1 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }