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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "White test 1" }}{PARA 0 "
" 0 "" {TEXT -1 53 "1.  a.  linear, variable coerricients, nonhomogene
ous" }}{PARA 0 "" 0 "" {TEXT -1 18 "     b.  same as a" }}{PARA 0 "" 
0 "" {TEXT -1 34 "     c.  nonlinear because of the " }{XPPEDIT 18 0 "
y^2" "6#*$%\"yG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 "2.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "restart:d2:=t*diff(y(t)
,t,t)+(1-t)*diff(y(t),t)-y(t)=0;\nd2s:=simplify(subs(y(t)=u(t)*exp(t),
d2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "### WARNING: `dso
lve` has been extensively rewritten, many new result forms can occur a
nd options are slightly different, see help page for details\ndsolve(d
2s,u(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ans2:=exp(t)*r
hs(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "3." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 61 "d3:=x^2*diff(y(x),x,x)-2*y(x)=0;\nsimplify(su
bs(y(x)=x^2,d3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simpli
fy(subs(y(x)=1/x,d3));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "3  b.  \+
Differentiate and sub x=1 in your head." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "solve(\{a+b=1, 2*a-b=1\},\{a,b\});" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 2 "4." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
223 "### WARNING: `dsolve` has been extensively rewritten, many new re
sult forms can occur and options are slightly different, see help page
 for details\ndsolve(\{diff(y(x),x,x)+2*diff(y(x),x)+10*y(x)=0, y(0)=0
, D(y)(0)=1\}, y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "5." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "d5:=diff(y(x),x,x,x)-3*diff(
y(x),x,x)-diff(y(x),x)+3*y(x)=1+exp(-x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "simplify(subs(y(x)=a+b*exp(-x),d5));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 23 "From this you see that " }{XPPEDIT 18 0 "
x*exp(-x)" "6#*&%\"xG\"\"\"-%$expG6#,$F$!\"\"F%" }{TEXT -1 11 " is nee
ded." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "simplify(subs(y(x)=
a+b*x*exp(-x),d5));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "So, " }
{XPPEDIT 18 0 "b=1/8" "6#/%\"bG*&\"\"\"\"\"\"\"\")!\"\"" }{TEXT -1 5 "
 and " }{XPPEDIT 18 0 "a=1/3" "6#/%\"aG*&\"\"\"\"\"\"\"\"$!\"\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 54 "Sol5:=1/8*x*exp(-x)+1/3: simplify(subs(y(x)=Sol5
,d5));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "6.  Using dsolve:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "s6:=\{diff(x(t),t)=2*x(t)-y
(t), diff(y(t),t)=x(t)-2*y(t)\};\n### WARNING: `dsolve` has been exten
sively rewritten, many new result forms can occur and options are slig
htly different, see help page for details\ndsolve(s6,\{x(t),y(t)\});" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The laplace option gives a more
 compact solution!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "### \+
WARNING: `dsolve` has been extensively rewritten, many new result form
s can occur and options are slightly different, see help page for deta
ils\ndsolve(s6,\{x(t),y(t)\},laplace);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 50 "You can get the 2nd order de in your head (almost)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "diff(diff(x(t),t)=2*x(t)-y(t
),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs(diff(y(t),t)=
x(t)-2*y(t), %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "d6:=sub
s(y(t)=2*x(t)-diff(x(t),t),%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 169 "### WARNING: `dsolve` has been extensively rewritten, many ne
w result forms can occur and options are slightly different, see help \+
page for details\nsd6:=dsolve(d6,x(t));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "y(t)=2*rhs(sd6)-diff(rhs(sd6),t);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 35 "A yellow test.  Problem  1 a.  is  " }{XPPEDIT 
18 0 "diff(y(t),t)=cos(t-y)" "6#/-%%diffG6$-%\"yG6#%\"tGF*-%$cosG6#,&F
*\"\"\"F(!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "1a.  no
nlinear" }}{PARA 0 "" 0 "" {TEXT -1 50 "1b.  linear, variable coeffici
ents, nonhomogeneous" }}{PARA 0 "" 0 "" {TEXT -1 30 "1c. nonlinear (be
cause of the " }{XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" }{TEXT -1 1 ")" }
}{PARA 0 "" 0 "" {TEXT -1 21 "2.  Same as #2 above." }}{PARA 0 "" 0 "
" {TEXT -1 20 "3.  Same as #3 above" }}{PARA 0 "" 0 "" {TEXT -1 2 "4.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "### WARNING: `dsolve` \+
has been extensively rewritten, many new result forms can occur and op
tions are slightly different, see help page for details\ndsolve(\{diff
(y(x),x,x)+2*diff(y(x),x)+5*y(x)=0, y(0)=0, D(y)(0)=1\}, y(x));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "5.  Either use dsolve (with the l
aplace option) or do it by hand (notice that all you need is a functio
n " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "
-diff(y(x),x)=cos(x)" "6#/,$-%%diffG6$-%\"yG6#%\"xGF+!\"\"-%$cosG6#F+
" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "subs(y(
x)=a*cos(x)+b*sin(x),diff(y(x),x,x)-diff(y(x),x)+y(x)=cos(x));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 4 "So, " }{XPPEDIT 18 0 "a=0" "6#/%\"aG\"\"!
" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "b=-1" "6#/%\"bG,$\"\"\"!\"\"" }
{TEXT -1 9 " works:  " }{XPPEDIT 18 0 "y=-sin(x)" "6#/%\"yG,$-%$sinG6#
%\"xG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 2 "6." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 231 "s6:=\{diff(x(t),t)=-2*x(t)+y(t), diff(y(t),t)=-
x(t)+2*y(t)\};\n### WARNING: `dsolve` has been extensively rewritten, \+
many new result forms can occur and options are slightly different, se
e help page for details\ndsolve(s6,\{x(t),y(t)\});" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 59 "The laplace option sometimes gives a more compact
 solution!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "### WARNING:
 `dsolve` has been extensively rewritten, many new result forms can oc
cur and options are slightly different, see help page for details\ndso
lve(s6,\{x(t),y(t)\},laplace);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 
"You can get the 2nd order de in your head (almost)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "diff(diff(x(t),t)=-2*x(t)+y(t),t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "subs(diff(y(t),t)=-x(t)+2*y(
t), %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "d6:=subs(y(t)=2*
x(t)+diff(x(t),t),%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "#
## WARNING: `dsolve` has been extensively rewritten, many new result f
orms can occur and options are slightly different, see help page for d
etails\nsd6:=dsolve(d6,x(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "y(t)=2*rhs(sd6)+diff(rhs(sd6),t);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "The other white test.  1a. is  " }{XPPEDIT 18 0 "diff(y(t
),t,t)=cos(t-t*y)" "6#/-%%diffG6%-%\"yG6#%\"tGF*F*-%$cosG6#,&F*\"\"\"*
&F*F/F(F/!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "1a.  no
nlinear" }}{PARA 0 "" 0 "" {TEXT -1 40 "1b.  linear, nonconstant, nonh
omogeneous" }}{PARA 0 "" 0 "" {TEXT -1 37 "1c.  linear, constant, nonh
omogeneous" }}{PARA 0 "" 0 "" {TEXT -1 18 "2.  Like #2 above." }}
{PARA 0 "" 0 "" {TEXT -1 26 "3. a.  Like #3a above   b." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "with(linalg): wronskian([x^2,1/x],x
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(%);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 105 "Because it is non zero, the set is a fun
damental solution.  Linearly dependent is also acceptable answer." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "4." }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "### WARNING: `dsolve` h
as been extensively rewritten, many new result forms can occur and opt
ions are slightly different, see help page for details\ndsolve(\{diff(
y(x),x,x)+2*diff(y(x),x)+10*y(x)=0,y(0)=0,D(y)(0)=2\},y(x));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "5.  Either use dsolve (with the l
aplace option) or do it by hand (notice that all you need is a functio
n " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "
-diff(y(x),x)=sin(x)" "6#/,$-%%diffG6$-%\"yG6#%\"xGF+!\"\"-%$sinG6#F+
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 73 "subs(y(x)=a*cos(x)+b*sin(x),3*diff(y(x),x,x)-
diff(y(x),x)+3*y(x)=sin(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 4 "So, " }{XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\"
" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "b=0" "6#/%\"bG\"\"!" }{TEXT -1 9 "
 works:  " }{XPPEDIT 18 0 "y=cos(x)" "6#/%\"yG-%$cosG6#%\"xG" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "6.  " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 229 "s6:=\{diff(x(t),t)=2*x(t)+y(t), diff(y(t),t)=x(t)+
2*y(t)\};\n### WARNING: `dsolve` has been extensively rewritten, many \+
new result forms can occur and options are slightly different, see hel
p page for details\ndsolve(s6,\{x(t),y(t)\});" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 59 "The laplace option sometimes gives a more compact solu
tion!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "### WARNING: `dso
lve` has been extensively rewritten, many new result forms can occur a
nd options are slightly different, see help page for details\ndsolve(s
6,\{x(t),y(t)\},laplace);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "You \+
can get the 2nd order de in your head (almost)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "diff(diff(x(t),t)=2*x(t)+y(t),t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs(diff(y(t),t)=x(t)+2*y(t), %);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "d6:=subs(y(t)=-2*x(t)+d
iff(x(t),t),%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "### WAR
NING: `dsolve` has been extensively rewritten, many new result forms c
an occur and options are slightly different, see help page for details
\nsd6:=dsolve(d6,x(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "
y(t)=-2*rhs(sd6)+diff(rhs(sd6),t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 31 "The other yellow test.  1a. is " }{XPPEDIT 18 0 "diff(y(t),t,t)
=cos(t)+y" "6#/-%%diffG6%-%\"yG6#%\"tGF*F*,&-%$cosG6#F*\"\"\"F(F/" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "1a.  linear, constant, n
onhomogeneous" }}{PARA 0 "" 0 "" {TEXT -1 14 "1b. nonlinear." }}{PARA 
0 "" 0 "" {TEXT -1 37 "1c.  linear, constant, nonhomogeneous" }}{PARA 
0 "" 0 "" {TEXT -1 21 "2.  Same as 2. above." }}{PARA 0 "" 0 "" {TEXT 
-1 2 "3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "det(wronskian([
x^2,1/x],x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve(\{a+
b=3, 2*a-b=0\}, \{a, b\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "s2=x^2+2/x;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "4." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "### WARNING: `dsolve` has been ext
ensively rewritten, many new result forms can occur and options are sl
ightly different, see help page for details\ndsolve(\{diff(y(x),x,x)+2
*diff(y(x),x)+10*y(x)=0,y(0)=2,D(y)(0)=0\},y(x));" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 21 "5.  Like the first 5." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 72 "d5:=2*diff(y(x),x,x,x)+3*diff(y(x),x,x)-2*diff(y(x)
,x)-3*y(x)=1+exp(-x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "si
mplify(subs(y(x)=a+b*exp(-x),d5));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 23 "From this you see that " }{XPPEDIT 18 0 "x*exp(-x)" "6#*&%\"xG
\"\"\"-%$expG6#,$F$!\"\"F%" }{TEXT -1 11 " is needed." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "simplify(subs(y(x)=a+b*x*exp(-x),d5
));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "So, " }{XPPEDIT 18 0 "b=-1/
2" "6#/%\"bG,$*&\"\"\"\"\"\"\"\"#!\"\"F*" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "a=-1/3" "6#/%\"aG,$*&\"\"\"\"\"\"\"\"$!\"\"F*" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "Sol5:=-1/2*x*exp(-x)-1/3: simplify(subs(y(x)=Sol5,d5)
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "6." }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "s6:=\{diff(
x(t),t)=-2*x(t)-y(t), diff(y(t),t)=-x(t)+2*y(t)\};\n### WARNING: `dsol
ve` has been extensively rewritten, many new result forms can occur an
d options are slightly different, see help page for details\ndsolve(s6
,\{x(t),y(t)\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The laplace o
ption sometimes gives a more compact solution!" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 179 "### WARNING: `dsolve` has been extensively re
written, many new result forms can occur and options are slightly diff
erent, see help page for details\ndsolve(s6,\{x(t),y(t)\},laplace);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "You can get the 2nd order de in \+
your head (almost)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "diff(
diff(x(t),t)=-2*x(t)-y(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "subs(diff(y(t),t)=-x(t)+2*y(t), %);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "d6:=subs(y(t)=-2*x(t)-diff(x(t),t),%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "### WARNING: `dsolve` has been ext
ensively rewritten, many new result forms can occur and options are sl
ightly different, see help page for details\nsd6:=dsolve(d6,x(t));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "y(t)=-2*rhs(sd6)-diff(rhs(s
d6),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
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