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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The pop test:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "x:=evalf(Pi/2);y:=1.;\nfor i to 10 \+
do\n  y:=y+0.1*(evalf(sin(x))+y);\n  x:=x+0.1;\nod:\ny;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "restart:dsolve(\{diff(y(x),x)-y(x)/
x=x^(-2),y(1)=2\},y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 
"y:=evalf(Pi/2);x:=1.;\nfor i to 10 do\n  y:=y+0.1*(evalf(sin(y))+x);
\n  x:=x+0.1;\nod:\ny;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "r
estart:dsolve(\{diff(y(x),x)-y(x)/x=x^2,y(1)=2\},y(x));;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "W
eek 6.  pp. 169-188" }}{PARA 0 "" 0 "" {TEXT -1 32 "Exercises.  p. 172
-  5, 6, 8, 10" }}{PARA 0 "" 0 "" {TEXT -1 27 "    p. 180-  11, 12, 19
, 21" }}{PARA 0 "" 0 "" {TEXT -1 59 "    p. 187  13, 22, 28, 29, 32, 3
3  In 29, the equation is " }{XPPEDIT 18 0 "diff(y(x),x,x)+13*diff(y(x
),x)+36*y(x)=0" "/,(-%%diffG6%-%\"yG6#%\"xGF*F*\"\"\"*&\"#8F+-F%6$-F(6
#F*F*F+F+*&\"#OF+-F(6#F*F+F+\"\"!" }{TEXT -1 33 ".  My book leaves out
 the last y." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "172 #6" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "d6:=x*diff(y(x),x,x)+(1-2*x)*diff(y
(x),x)+(x-1)*y(x)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dso
lve(d6,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "d6s:=subs(
y(x)=v(x)*exp(x),d6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "d6
t:=simplify(d6s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve
(d6t,v(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "#8" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "d8:=x*diff(y(x),x,x,x)+(1-x)*diff(y
(x),x,x)+x*diff(y(x),x)-y(x)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "dsolve(d8,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 
"d8s:=subs(y(x)=x*v(x),d8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "d8ss:=simplify(d8s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 
"dsolve(d8ss,v(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "#9" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "d9:=diff(y(x),x,x)-x*diff(y(
x),x)+y(x)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(y(x)=
x*v(x),d9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\")
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "int(exp(x^2/2)/x^2,x);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "#10" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 54 "d10:=(1-x^2)*diff(y(x),x,x)-2*x*diff(y(x),x)+2*y(x
)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "d10s:=simplify(subs
(y(x)=x*v(x),d10));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(diff(v(
x),x)=u(x),d10s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "int((4
*x^2-2)/(x-x^3),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "exp(
\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(\",x);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 4 "\"*x;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "subs(y(x)=\",d10);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "x*rhs(dsolve(d10s,v(x)));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "dsolve(d10,y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 72 "The equation in 9 is somewhat like van der Pol's equation  (see
 p. 315):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "vdp:=diff(y(x)
,x,x)-(1-y(x)^2)*diff(y(x),x)+y(x)=0;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "dsolve(vdp,y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 48 "So Maple can't do it exactly.  (I can't either.)" }}{PARA 0 "" 
0 "" {TEXT -1 29 "p. 180 #11  General solution." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 100 "restart:de:=a*diff(y(x),x,x)+b*diff(y(x),x)+c
*y(x);\nsubs(y(x)=exp(r*x),de); simplify(\"); solve(\",r);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "restart: d11:=4*diff(w(t),t,t)+20*d
iff(w(t),t)+25*w(t)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "s
implify(subs(w(t)=v(t)*exp(-5*t/2),d11));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "dsolve(d11,w(t));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "factor(4*r^2+20*r+25); solve(4*r^2+20*r+25,r);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "So it is a double root." }}{PARA 
0 "" 0 "" {TEXT -1 10 "p. 180 #12" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "d12:=3*diff(y(x),x,x)+11*diff(y(x),x)-7*y(x)=0;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(d12,y(x));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "solve(3*r^2+11*r-7,r);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 7 "180 #19" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "d19:
=diff(y(x),x,x)-4*diff(y(x),x)-5*y(x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "dsolve(d19,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "solve(r^2-4*r-5,r);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "180 #21" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "d21:=a*diff(y(x),x)+b*y(x)=0;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 40 "d21s:=simplify(subs(y(x)=exp(r*x),d21));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(d21s,r);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "The solution is just " }{XPPEDIT 
18 0 "C" "I\"CG6\"" }{XPPEDIT 18 0 "exp(-b/a*t)" "-%$expG6#,$*(%\"bG\"
\"\"%\"aG!\"\"%\"tGF(F*" }{TEXT -1 7 ".  No, " }{XPPEDIT 18 0 "x" "I\"
xG6\"" }{TEXT -1 6 ", not " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 1 
"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(d21,y(x));" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "p. 187 # 13, 22, 28, 29, 32, 33" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "d13:=diff(y(x),x,x)+2*dif
f(y(x),x)+5*y(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve
(d13,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(r^2+2*
r+5,r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "d22:=diff(y(x),x
,x)+2*diff(y(x),x)+17*y(x); i22:=y(0)=1, D(y)(0)=-1;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 23 "dsolve(\{d22,i22\},y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(d22,y(x));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 50 "restart:d28:=diff(y(x),x,x)+b*diff(y(x),x
)+4*y(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "i28:=y(0)=0,D(
y)(0)=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "b:=2; s282:=dso
lve(\{d28,i28\},y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "p
2:=plot(rhs(s282),x=0..4,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "b:=4; s282:=dsolve(\{d28,i28\},y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "p4:=plot(rhs(s282),x=0..4,color=gre
en):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "b:=5; s282:=dsolve(
\{d28,i28\},y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "p5:=p
lot(rhs(s282),x=0..4,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "b:=1/10; s282:=dsolve(\{d28,i28\},y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "pt:=plot(rhs(s282),x=0..4,color=pur
ple):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "b:=0; s282:=dsolve(\{d28,i28\},y(x));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "pz:=plot(rhs(s282),x=0..4,c
olor=orange):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " s282:=dso
lve(\{d28,i28\},y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "w
ith(plots):display([p2,p4,p5,pt,pz]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 79 "solve(r^2+5*r+4,r);solve(r^2+4*r+4,r);solve(r^2+2*r+4
,r);solve(r^2+1/10*r+4,r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "b:='b'; solve(r^2+b*r+4,r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "with(linalg);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a:
=vector([cos(2*x),sin(2*x)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "wronskian(a,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det
(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}}
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