{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 256 11 "Math 2250-3" }}{PARA 257 "" 0 "" {TEXT 257 49 "Notes for Separable Differential Equations - %1.4" }} {PARA 258 "" 0 "" {TEXT -1 27 "Wednesday September 1, 2004" }}{PARA 0 "" 0 "" {TEXT -1 23 "A differential equation" }}{EXCHG {PARA 11 "" 1 " " {XPPMATH 20 "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"hG6$%\"xG%\"yG" }}}{PARA 0 "" 0 "" {TEXT -1 10 "is called " }{TEXT 258 9 "separable" }{TEXT 260 1 " " }{TEXT -1 65 "iff h(x,y) is a product of a function of x tim es a function of y," }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#dyG \"\"\"%#dxG!\"\"*&-%\"gG6#%\"xGF&-%$phiG6#%\"yGF&" }}}{PARA 0 "" 0 "" {TEXT -1 28 "This is equivalent to the DE" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%\"gG6#%\"xGF&-%\"fG6#%\"yGF (" }}}{PARA 0 "" 0 "" {TEXT -1 41 "where f and phi are reciprocal func tions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 13 "How to solve:" }}{PARA 0 "" 0 "" {TEXT -1 131 "The algorithm is very simple, but magi c: treat dy/dx as a quotient of differntials (?!), and multiply throu gh to rewrite the DE as" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&- %\"fG6#%\"yG\"\"\"%#dyGF)*&-%\"gG6#%\"xGF)%#dxGF)" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 93 "Then antidifferentiate the left side with respect \+ to y and the right side with respect to x. " }}}{EXCHG {PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$-%\"fG6#%\"yGF*,&-F%6$-%\"gG6#%\"xGF1\"\"\" %\"CGF2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "If F(y) and G(x) are a ntiderivatives of f(y) and g(x), respectively, then this is the soluti on" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"yG,&-%\"GG6# %\"xG\"\"\"%\"CGF-" }}}{PARA 0 "" 0 "" {TEXT -1 177 "This equation def ines y implicitly as a function of x. Sometimes you can use algebra t o explicitly solve for y. The constant C can be adjusted to solve ini tial value problems." }}{PARA 0 "" 0 "" {TEXT 261 21 "Why the method w orks:" }}{PARA 0 "" 0 "" {TEXT -1 146 "The use of differentials is dis guising an application of the chain rule. Here is the explanation for the magic method: The differential equation" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%\"gG6#%\"xGF&-%\"fG6#%\" yGF(" }}}{PARA 0 "" 0 "" {TEXT -1 19 "CAN be rewritten as" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'deqtn2G/*&-%\"fG6#%\"yG\"\"\"7#*&%# dyGF+%#dxG!\"\"F+-%\"gG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 61 "If y(x ) is any solution to deqtn2, then the left side, namely" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%\"fG6#-%\"yG6#%\"xG\"\"\"7#*&%#dyG F+%#dxG!\"\"F+" }}}{PARA 0 "" 0 "" {TEXT -1 38 "is the derivative with respect to x of" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"FG6#-% \"yG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 229 "whenever F(y) is an anti derivative of f(y) (with respect to y). This is just the chain rule! \+ Thus if G(x) is any antiderivative of g(x) (w.r.t.x), we can legally \+ antidifferentiate deqtn2 with respect to x (on both sides) to get" }} {EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#-%\"yG6#%\"xG,&-%\"GGF )\"\"\"%\"CGF." }}}{PARA 0 "" 0 "" {TEXT -1 37 "which is what we got b y magic before!" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 17 "Example 1 page 32" }{TEXT -1 18 ": We wish to solve" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&%\" yGF&!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"!\"\"(" }}} {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 solution doesn't actually give us the solution y(x)=0. Solutions which exist to separable DE's whic h are in addition to the ones we get are called " }{TEXT 263 21 "\"sin gular solutions.\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 20 "slo pe field picture:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "restart :with(plots):with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 294 "deqtn:=diff(y(x),x)=-6*x*y(x): #this is example 2\ndsolve(\{deqt n,y(0)=7\},y(x)); #Maple solution\nDEplot(deqtn,y(x),x=-2..2,\{[y(0)= 7],[y(0)=-4]\},y=-10..10,arrows=line, color=black,linecolor=black,dir grid=[30,30],stepsize=.1,\n title=`Figure 1.4.1 page 31`); #slope fiel d with two solution graphs" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6 #%\"xG,$*&\"\"(\"\"\"-%$expG6#,$*&\"\"$F+)F'\"\"#F+!\"\"F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 375 229 229 {PLOTDATA 2 "6(-%'CURVESG6acn7$7$$!3& e6\\?2rG+#!#<$!3E&31TNral*F*7$$!39%)3&z#*Gr*>F*$!3[\"R*ekGXM5!#;7$7$$! 3fU[5*H_^'=F*$!30T9(H@j&)*e=F*$!3!f&Gqy#[W.\"F27$7$$!3 58ylxtYF_6T%[33s\"F*$!3SKeF^DWM5F27$7$$!3,bL=& 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