{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 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 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "1.4  #8" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "de:=diff(y(x),x)=1-sin(y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dsolve(\{de,y(0)=0\},y(x));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(y(x)=rhs(\"),de);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(\",trig);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "2*sin(arctan(x/(x+2)))*cos(arctan(x
/(x+2)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(1-\")
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "1.4  #12  Refer to Example 2
.   It refers to the IVP " }{XPPEDIT 18 0 "diff(y(t),t)=y(t), y(0)=1" 
"6$/-%%diffG6$-%\"yG6#%\"tGF*-F(6#F*/-F(6#\"\"!\"\"\"" }}{PARA 0 "" 0 
"" {TEXT -1 32 "The approximation with stepsize " }{XPPEDIT 18 0 "1/n
" "*&\"\"\"\"\"\"%\"nG!\"\"" }{TEXT -1 7 " gives " }{XPPEDIT 18 0 "y[n
]=(1+1/n)^n" "/&%\"yG6#%\"nG),&\"\"\"\"\"\"*&\"\"\"F*F&!\"\"F*F&" }
{TEXT -1 36 ".  (I'll do this on the whiteboard.)" }{MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "E:=exp(1): limit((1+1/n)^
n,n=infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "\"-E;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "1.4, #13  " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 38 "limit((E-(1+1/n)^n)/(1/n),n=infinity);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Problems for you:  " }}{PARA 0 "" 
0 "" {TEXT -1 19 "1.4           5, 12" }}{PARA 0 "" 0 "" {TEXT -1 33 "
2.1, 2.2    3, 12, 23, 27, 29, 32" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "p. 44 #5 -- at whiteboard.  Is " }
{XPPEDIT 18 0 "s^2+diff(s(t),t)=(s+1)/(s*t)" "/,&*$%\"sG\"\"#\"\"\"-%%
diffG6$-F%6#%\"tGF-F'*&,&F%F'\"\"\"F'F'*&F%F'F-F'!\"\"" }{TEXT -1 11 "
 separable?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 11 "#15  Solve " }{XPPEDIT 18 0 "dy(x)/y(x) +y(x)*exp(cos(x))*sin(x
)*dx=0." "/,&*&-%#dyG6#%\"xG\"\"\"-%\"yG6#F(!\"\"F)**-F+6#F(F)-%$expG6
#-%$cosG6#F(F)-%$sinG6#F(F)%#dxGF)F)$\"\"!F<" }}{PARA 0 "" 0 "" {TEXT 
-1 18 "This is separable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 11 "#26  Solve " }{XPPEDIT 18 0 "sqrt(y(x))*dx+(1+x)*d
y=0" "/,&*&-%%sqrtG6#-%\"yG6#%\"xG\"\"\"%#dxGF,F,*&,&\"\"\"F,F+F,F,%#d
yGF,F,\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "So is this
 one." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "
#28  Sketch solution:  " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "diff(y(t),t)
=2*y(t)-2*y(t)*t, y(0)=3" "6$/-%%diffG6$-%\"yG6#%\"tGF*,&*&\"\"#\"\"\"
-F(6#F*F.F.*(\"\"#F.-F(6#F*F.F*F.!\"\"/-F(6#\"\"!\"\"$" }{TEXT -1 1 ".
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "ivp:=\{diff(y(t),t)=2*y(t)*(1-t),y(0)=3\};" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(ivp,y(t));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(rhs(\"),t=0..3);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "# 31  Solve " }{XPPEDIT 18 0 "diff
(y(x),x)=(x-3)*(y(x)+1)^(2/3)" "/-%%diffG6$-%\"yG6#%\"xGF)*&,&F)\"\"\"
\"\"$!\"\"F,),&-F'6#F)F,\"\"\"F,*&\"\"#F,\"\"$F.F," }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 39 "d31:=diff(y(x),x)=(x-3)*(y(x)+1)^(2/3);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(d31,y(x));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Obviously, " }{TEXT -1 1 " " }
{XPPEDIT 18 0 "y(x)=-1 " "/-%\"yG6#%\"xG,$\"\"\"!\"\"" }{TEXT -1 1 " \+
" }{TEXT -1 35 "satisfies the equation.   How come?" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "# 33  Mixing.  \+
On whiteboard and later (in Section 3.2)." }}}}{MARK "17 0 0" 17 }
{VIEWOPTS 1 1 0 2 1 1805 }