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{SECT 0 {PARA 256 "" 0 "" {TEXT 256 11 "Math 2280-1" }}{PARA 257 "" 0 
"" {TEXT 257 49 "Notes for Separable Differential Equations - %1.4" }}
{PARA 258 "" 0 "" {TEXT -1 23 "Friday January 13, 2006" }}{PARA 0 "" 
0 "" {TEXT -1 60 "As we've already discussed in class, a differential \+
equation" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "dy/dx = h(x,y);" "6#/*&%#
dyG\"\"\"%#dxG!\"\"-%\"hG6$%\"xG%\"yG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 10 "is called " }{TEXT 258 9 "separable" }{TEXT 260 1 " " }
{TEXT -1 65 "iff h(x,y) is a product of a function of x times a functi
on of y," }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "dy/dx = g(x)*phi(y);" "6#
/*&%#dyG\"\"\"%#dxG!\"\"*&-%\"gG6#%\"xGF&-%$phiG6#%\"yGF&" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "This is equivalent to the DE" }}
{PARA 261 "" 0 "" {XPPEDIT 18 0 "dy/dx = g(x)/f(y);" "6#/*&%#dyG\"\"\"
%#dxG!\"\"*&-%\"gG6#%\"xGF&-%\"fG6#%\"yGF(" }{TEXT -1 1 "," }}{PARA 0 
"" 0 "" {TEXT -1 41 "where f and phi are reciprocal functions." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 13 "How to solve:" }}{PARA 0 
"" 0 "" {TEXT -1 131 "The algorithm is very simple, but magic:  treat \+
dy/dx as a quotient of differntials (?!), and multiply through to rewr
ite the DE as" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "f(y)*dy = g(x)*dx;" 
"6#/*&-%\"fG6#%\"yG\"\"\"%#dyGF)*&-%\"gG6#%\"xGF)%#dxGF)" }{TEXT -1 1 
"." }}{PARA 0 "" 0 "" {TEXT -1 93 "Then antidifferentiate the left sid
e with respect to y and the right side with respect to x. " }}{PARA 
263 "" 0 "" {XPPEDIT 18 0 "Int(f(y),y) = Int(g(x),x)+C;" "6#/-%$IntG6$
-%\"fG6#%\"yGF*,&-F%6$-%\"gG6#%\"xGF1\"\"\"%\"CGF2" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 103 "If F(y) and G(x) are antiderivatives of \+
f(y) and g(x), respectively, then this is the implicit solution" }}
{PARA 264 "" 0 "" {XPPEDIT 18 0 "F(y) = G(x)+C;" "6#/-%\"FG6#%\"yG,&-%
\"GG6#%\"xG\"\"\"%\"CGF-" }{TEXT -1 1 "." }}{PARA 265 "" 1 "" {TEXT 
-1 177 "This equation defines y implicitly as a function of x.  Someti
mes you can use algebra to explicitly solve for y.  The constant C can
 be adjusted to solve initial value problems." }}{PARA 0 "" 0 "" 
{TEXT 261 21 "Why the method works:" }}{PARA 0 "" 0 "" {TEXT -1 146 "T
he use of differentials is disguising an application of the chain rule
.  Here is the explanation for the magic method:  The differential equ
ation" }}{PARA 266 "" 0 "" {XPPEDIT 18 0 "dy/dx = g(x)/f(y);" "6#/*&%#
dyG\"\"\"%#dxG!\"\"*&-%\"gG6#%\"xGF&-%\"fG6#%\"yGF(" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 19 "CAN be rewritten as" }}{PARA 267 "" 0 "" 
{XPPEDIT 18 0 "deqtn2 := f(y)*[dy/dx] = g(x);" "6#>%'deqtn2G/*&-%\"fG6
#%\"yG\"\"\"7#*&%#dyGF+%#dxG!\"\"F+-%\"gG6#%\"xG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 61 "If y(x) is any solution to deqtn2, then t
he left side, namely" }}{PARA 268 "" 0 "" {XPPEDIT 18 0 "f(y(x))*[dy/d
x];" "6#*&-%\"fG6#-%\"yG6#%\"xG\"\"\"7#*&%#dyGF+%#dxG!\"\"F+" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "is the derivative with respect \+
to x of" }}{PARA 269 "" 0 "" {XPPEDIT 18 0 "F(y(x));" "6#-%\"FG6#-%\"y
G6#%\"xG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 229 "whenever F(y)
 is an antiderivative of f(y) (with respect to y).  This is just the c
hain rule!  Thus if G(x) is any antiderivative of g(x) (w.r.t.x), we c
an legally antidifferentiate deqtn2 with respect to x (on both sides) \+
to get" }}{PARA 270 "" 0 "" {XPPEDIT 18 0 "F(y(x)) = G(x)+C;" "6#/-%\"
FG6#-%\"yG6#%\"xG,&-%\"GGF)\"\"\"%\"CGF." }{TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 37 "which is what we got by magic before!" }}{PAGEBK }
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 17 "Example 1 page 32" }
{TEXT -1 18 ": We wish to solve" }}{PARA 271 "" 0 "" {XPPEDIT 18 0 "dy
/dx = -6*x*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&%\"yGF&!\"'" }
{TEXT -1 0 "" }}{PARA 272 "" 0 "" {XPPEDIT 18 0 "y(0) = 7;" "6#/-%\"yG
6#\"\"!\"\"(" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Work:" }}
{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 "" }}{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 -1 183 "Notice, our method for the general s
olution doesn't actually give us the solution y(x)=0.  Solutions which
 exist to separable DE's which are in addition to the ones we get are \+
called " }{TEXT 263 21 "\"singular solutions.\"" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 265 20 "slope field picture:" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 34 "restart:with(plots):with(DEtools):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 335 "deqtn:=diff(y(x),x)=-6*x*y(
x):  #this is example 1 dsolve(\{deqtn,y(0)=7\},y(x));  #Maple solutio
n\nDEplot(deqtn,y(x),x=-2..2,\{[y(0)=7],[y(0)=-4],\n[y(0)=1],[y(0)=4],
[y(0)=-7],[y(0)=-1.5]\},y=-10..10,arrows=line,  color=black,linecolor=
black,dirgrid=[30,30],stepsize=.1,\n title=`Figure 1.4.1 page 32`); #s
lope field with two solution graphs" }}{PARA 13 "" 1 "" {GLPLOT2D 375 
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FechlFd]jl7$FhchlFa]jl7$F[dhlF^]jl7$F^dhlF[]jl7$FadhlFh\\jl7$FddhlFe\\
jl7$FgdhlFb\\jl7$FjdhlF_\\jl7$F]ehlF\\\\jl7$F`ehlFi[jl7$FcehlFf[jl7$Ff
ehlFc[jl7$FiehlF`[jl7$F\\fhlF][jl-%&STYLEG6#%%LINEG-%*THICKNESSG6#Fjbh
lFi[hl-%%VIEWG6$;$!+LLLL@Ff_hl$\"+LLLL@Ff_hl;$!+nmmm5F[`hl$\"+nmmm5F[`
hl-%+AXESLABELSG6$Q\"x6\"Q%y(x)Ffajl-%&TITLEG6#%5Figure~1.4.1~page~32G
" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C
urve 2" }}}}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 15 "Exam
ple extra: " }{TEXT -1 105 "(We discussed this DE, but not this initia
l value, when we talked about the existence uniqueness theorem)" }}
{PARA 273 "" 0 "" {XPPEDIT 18 0 "dy/dx = y^(2/3);" "6#/*&%#dyG\"\"\"%#
dxG!\"\"*$)%\"yG#\"\"#\"\"$F&" }{TEXT -1 0 "" }}{PARA 275 "" 0 "" 
{XPPEDIT 18 0 "y(-3) = -1;" "6#/-%\"yG6#!\"$!\"\"" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 40 "Solution from separation of variables:  \+
" }}{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 "" }}{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 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 216 "deqtn:=diff(y(x),x)=abs(y(x))^(2/3.0):\nDEplot(deqtn
,y(x),x=-4..4,\{[y(-3)=-1]\},y=-5..5,arrows=line,   color=black,lineco
lor=black,dirgrid=[30,30],stepsize=.1,\n title=`how many solutions rea
lly?`); #Maple missed a few " }}{PARA 13 "" 1 "" {GLPLOT2D 360 294 
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