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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Week 3  46-65  Exercises p
. 52-  10, 14, 28, 29, 30" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 35 "10.  Find the general solution of  " }{XPPEDIT 18 
0 "x*diff(y(x),x)+2*y(x)=x^(-3)" "/,&*&%\"xG\"\"\"-%%diffG6$-%\"yG6#F%
F%F&F&*&\"\"#F&-F+6#F%F&F&)F%,$\"\"$!\"\"" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "d10:=x*diff(y(x),x)+2*y(x)=x^(-3);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(d10,y(x));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "14
.  Same for " }{XPPEDIT 18 0 "x*diff(y(x),x)+3*y(x)+2*x^2=x^3+4*x" "/,
(*&%\"xG\"\"\"-%%diffG6$-%\"yG6#F%F%F&F&*&\"\"$F&-F+6#F%F&F&*&\"\"#F&*
$F%\"\"#F&F&,&*$F%\"\"$F&*&\"\"%F&F%F&F&" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "d14:= x*diff(y(x),x)+3*y(x)+2*x^2=x
^3+4*x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(d14,y(x))
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
16 "28.a  Show that " }{XPPEDIT 18 0 "y=exp(-x)" "/%\"yG-%$expG6#,$%\"
xG!\"\"" }{TEXT -1 38 " is a solution of the linear equation " }
{XPPEDIT 18 0 "diff(y(x),x)+y(x)=0" "/,&-%%diffG6$-%\"yG6#%\"xGF*\"\"
\"-F(6#F*F+\"\"!" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "dsolve(diff(y(x),x)+y(x)=0,y(x));" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 42 "    .b  Show a constant times this is too." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 54 "    .c  Show that for any linear equation of the form " }
{XPPEDIT 18 0 "diff(y(x),x)+P(x)*y(x)=0" "/,&-%%diffG6$-%\"yG6#%\"xGF*
\"\"\"*&-%\"PG6#F*F+-F(6#F*F+F+\"\"!" }{TEXT -1 5 ", if " }{XPPEDIT 
18 0 "y(x) " "-%\"yG6#%\"xG" }{TEXT -1 39 " is a solution then any con
stant times " }{XPPEDIT 18 0 "y(x)" "-%\"yG6#%\"xG" }{TEXT -1 15 " is \+
a solution." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dsolve(diff(
y(x),x)+P(x)*y(x)=0,y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "29.
  Use your ingenuity to solve the equation " }{XPPEDIT 18 0 "diff(y(x)
,x)=1/(exp(4*y(x))+2*x)" "/-%%diffG6$-%\"yG6#%\"xGF)*&\"\"\"\"\"\",&-%
$expG6#*&\"\"%F,-F'6#F)F,F,*&\"\"#F,F)F,F,!\"\"" }{TEXT -1 1 "." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "d29:=diff(y(x),x)=1/(exp(4*y
(x))+2*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(d29,y(
x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(\",x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x=(_C1+(1/2)*exp(2*y(x)))*ex
p(2*y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "30.  The equation " }
{XPPEDIT 18 0 "diff(y(x),x)+2*y(x)=x*(y(x))^(-2)" "/,&-%%diffG6$-%\"yG
6#%\"xGF*\"\"\"*&\"\"#F+-F(6#F*F+F+*&F*F+)-F(6#F*,$\"\"#!\"\"F+" }
{TEXT -1 40 " is a Bernoulli equation (more later).  " }}{PARA 0 "" 0 
"" {TEXT -1 30 "    .a  Show the substitution " }{XPPEDIT 18 0 "v=y^3
" "/%\"vG*$%\"yG\"\"$" }{TEXT -1 25 " gives a linear equation." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "d30:=diff(y(x),x)+2*y(x)=x*(
y(x))^(-2);\nsubs(y(x)=v(x)^(1/3),d30);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "da:=simplify(\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 42 "    .b  Solve this equation and then find " }{XPPEDIT 18 0 "y(x
)" "-%\"yG6#%\"xG" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "db:=simplify(v(x)^(2/3)*\");" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "dsolve(db,v(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "dsolve(d30,y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 92 "Note that Maple knows a formula for the general solution to the
 first order linear equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 41 "dsolve(diff(y(x),x)+P(x)*y(x)=Q(x),y(x));" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 70 "Here is something to watch for.  You sometimes get r
eally crazy plots." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "poly:
=10^(16)*expand((1-x)^3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "plot(poly,x=1..1.000001,numpoints=20);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 25 "Now change the precision:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "Digits:=30; plot(poly,x=1..1.000001,numpoints=20);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "2.4  Exact equations.  p. 63-  1
1, 14, 21, 24, 32." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 29 "32.  Orthogonal trajectories." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 66 "with(plots);\nimplicitplot(\{x^2 + y^2 = 1,x = y\},
x=-1..1,y=-1..1);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "implicitplot(\{x*y = .1,y^2-x^2=.1
,x*y = -.1,y^2-x^2=-.1,x*y = .05,y^2-x^2=.05,x*y = .05,y^2-x^2=-.05,x*
y = .08,y^2-x^2=-.08\},x=-1..1,y=-1..1);\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 73 "contourplot(\{cos(x*y)\},x=-6..6,y=-6..6,filled=tru
e,coloring=[red,green]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 
"contourplot(\{cos(x)*cos(y)\},x=-4*Pi..4*Pi,y=-4*Pi..4*Pi,filled=true
,coloring=[blue,green]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 
"F:=(x,y)->x*y-x^2+y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "or
thtraj:=proc(x1,x2,y1,y2)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "local \+
x,y,de;\nglobal F;\nwith(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 
"de:=diff(y(x),x)=diff(F(x,y),y)/diff(F(x,y),x);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "de:=subs(y=y(x),de);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "dsolve(de,y(x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(\",
_C1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(y(x)=y,\");" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 67 "contourplot(\{F(x,y),\"\},x=x1..x2,y=y1..
y2,coloring=[red,blue]);\nend;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "orthtraj(-2,2,-2,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "solve(y=c*x,c);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "de
:=diff(y(x),x)=diff(F(x,y),y)/diff(F(x,y),x);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "de:=subs(y=y(x),de);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "dsolve(de,y(x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(\",
_C1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(y(x)=y,\");" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(de,y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(\",_C1);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 15 "subs(y(x)=y,\");" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 31 "contourplot(\",x=-1..1,y=-1..1);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "46 0 0" 0 }{VIEWOPTS 1 1 0 1 
1 1803 }