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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Week 8  pp. 207-214, pp. 2
40-260" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 
"4.10  Systems of first order equations.  Exercises p. 214-  1, 3, 4, \+
5, 27, 29, 32." }}{PARA 0 "" 0 "" {TEXT -1 54 "# 2  Convert to a syste
m of two first order equations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 68 "restart: sys2:=\{diff(v(t),t)=cos(t-y(t))+y(t)^2, diff(y(t),t)
=v(t)\};" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 " # 6  Use elimination
 to find a general solution.  " }{XPPEDIT 18 0 "diff(x(t),t)=3*y(t)" "
/-%%diffG6$-%\"xG6#%\"tGF)*&\"\"$\"\"\"-%\"yG6#F)F," }{TEXT -1 2 ", " 
}{XPPEDIT 18 0 "diff(y(t),t)=2*x(t)-y(t)" "/-%%diffG6$-%\"yG6#%\"tGF),
&*&\"\"#\"\"\"-%\"xG6#F)F-F--F'6#F)!\"\"" }{TEXT -1 64 ".   I would di
fferentiate the second equation and plug in for x'" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 41 "d6:=diff(y(t),t,t)=2*3*y(t)-diff(y(t),t);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(d6,y(t));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "# 7  " }{XPPEDIT 18 0 "diff(x(t),t)
+2*y(t)=0" "/,&-%%diffG6$-%\"xG6#%\"tGF*\"\"\"*&\"\"#F+-%\"yG6#F*F+F+
\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "diff(x(t),t)-diff(y(t),t)=0.  \+
" "/,&-%%diffG6$-%\"xG6#%\"tGF*\"\"\"-F%6$-%\"yG6#F*F*!\"\"$\"\"!F3" }
{TEXT -1 10 "  Plug in " }{XPPEDIT 18 0 "-2*y(t)" ",$*&\"\"#\"\"\"-%\"
yG6#%\"tGF%!\"\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "d7:=-2*y(t)-diff(y(t),t)=0;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "dsolve(d7,y(t));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "dsolve(diff(x(t),t)+rhs(\")*2=0,x(t));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 38 "# 28  Same for a third order system.  " }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "sys28:=\{diff(x(t),t)=3*x
(t)+y(t)-z(t), diff(y(t),t)=x(t)+2*y(t)-z(t), diff(z(t),t)=3*x(t)+3*y(
t)-z(t)\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "s28a:=simplif
y(subs(\{sys28[2],sys28[3]\},diff(sys28[1],t)));" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 44 "s28b:=simplify(subs(sys28[2],diff(s28a,t)));
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Notice that 2y-z = y'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "subs(-2*y(t)+z(t)=-diff(y(t)
,t),s28b);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Didn't work!  Do it
 by hand, sort of." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "s28c:
=s28b+sys28[2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Now plug in fo
r y' from s28a's derivative." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "s28d:=solve(diff(s28a,t),diff(y(t),t));" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "eq28:=subs(diff(y(t),t)=s28d,s28c);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 44 "Rats!  It is nice that 0=0.  Start all ov
er." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "sys28;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sa:=diff(sys28[1],t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "sb:=simplify(subs(\{sys28[2],sys28[
3]\},sa));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "sc:=diff(sb,t
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sd:=subs(sys28[2],sc)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 68 "5.1 - 5.3  Exercises  p. 246 6, 8;  p. 253 5, 7, 10;
 p. 259 2, 3, 9." }}}}{MARK "30 0 0" 4 }{VIEWOPTS 1 1 0 1 1 1803 }