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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Week 7 pp. 188-207." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 208 "This wee
k we learn how to solve nonhomogeneous second order equations with con
stant coefficients.  We start with a little theory which is not restri
cted to constant coefficients but just depends on linearity." }}{PARA 
0 "" 0 "" {TEXT -1 37 "Exercises: p. 191- 4, 11, 18, 19, 20." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "d4:=diff(x(t),t,t)-4*diff(x(
t),t)+3*x(t)=-2*exp(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "
subs(x(t)=t*exp(t),d4); simplify(\");" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "dsolve(d4,x(t));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "d18:=diff(y(x),x,x)-diff(y(x),x)+y(x)=0; dsolve(d18,y
(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "restart: d18b:=dif
f(y(x),x,x)-diff(y(x),x)+y(x)=sin(x)-3*exp(2*x); s18b:=dsolve(d18b,y(x
));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Why can't Maple do this on
e like we do?  Try taking the real part." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "Re(\"); # you have to assume(x,real) first!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 46 "p. 200- 1, 17, 31, 32, 36, 41, 51, 56, 64
, 65." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "with(linalg):w:=ve
ctor([exp(-x)*cos(sqrt(2)*x),exp(-x)*sin(sqrt(2)*x)]);\ndet(wronskian(
w,x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "dsolve(diff(y(x),
x,x)+2*diff(y(x),x)+3*y(x)=sin(2*x),y(x),laplace);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 726 "itis:=-7/68*cos(2^(1/2)*x)*cos((2^(1/2)-2)
*x)*2^(1/2)+7/68*cos(2^(1/2)*x)*cos((2+2^(1/2))*x)*2^(1/2)+1/34*cos(2^
(1/2)*x)*sin((2^(1/2)-2)*x)-1/34*cos(2^(1/2)*x)*sin((2+2^(1/2))*x)-2/1
7*sin(2^(1/2)*x)*sin((2^(1/2)-2)*x)-2/17*cos(2^(1/2)*x)*cos((2^(1/2)-2
)*x)-1/34*sin(2^(1/2)*x)*cos((2^(1/2)-2)*x)-2/17*sin(2^(1/2)*x)*sin((2
+2^(1/2))*x)-2/17*cos(2^(1/2)*x)*cos((2+2^(1/2))*x)+3/34*cos(2^(1/2)*x
)*sin((2^(1/2)-2)*x)*2^(1/2)+7/68*sin(2^(1/2)*x)*sin((2+2^(1/2))*x)*2^
(1/2)-3/34*sin(2^(1/2)*x)*cos((2+2^(1/2))*x)*2^(1/2)+3/34*cos(2^(1/2)*
x)*sin((2+2^(1/2))*x)*2^(1/2)-7/68*sin(2^(1/2)*x)*sin((2^(1/2)-2)*x)*2
^(1/2)-3/34*sin(2^(1/2)*x)*cos((2^(1/2)-2)*x)*2^(1/2)+1/34*sin(2^(1/2)
*x)*cos((2+2^(1/2))*x)-(-4/17*cos(2*x)-1/17*sin(2*x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(itis,x=0..Pi);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(itis);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 67 "dsolve(diff(y(x),x,x)+2*diff(y(x),x)+3*y(x)
=sin(2*x),y(x),laplace);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 
"dsolve(diff(y(x),x,x)+2*diff(y(x),x)+3*y(x)=0,y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "int(exp(x)*sin(sqrt(2)*x)*sin(2*x),
x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(\{y(0)=0,D(y)(0
)=0\},\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "# 1  This is too ea
sy.  Isn't it clear that " }{XPPEDIT 18 0 "y=-3" "/%\"yG,$\"\"$!\"\"" 
}{TEXT -1 7 " works?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dso
lve(diff(y(x),x,x)+3*y(x)=9,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "diff(tan(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "diff(\",x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "#
 17  This one is also easy because the second derivative of " }
{XPPEDIT 18 0 "-11*x+1" ",&*&\"#6\"\"\"%\"xGF%!\"\"\"\"\"F%" }{TEXT 
-1 9 " is zero." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart:
dsolve(diff(y(x),x,x)-y(x)=-11*x+1,y(x));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 210 "# 31  Of course, dsolve does this one.  It might be fun \+
to see how many different ways you can come up with a solution.  It is
   linear first order   variables separable   the homogeneous equation
 succumbs to  " }{XPPEDIT 18 0 "y=exp(r*x)" "/%\"yG-%$expG6#*&%\"rG\"
\"\"%\"xGF)" }{TEXT -1 32 " sub.  Plus Maple can do it too." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "dsolve(\{diff(y(x),x)-y(x)=1
,y(0)=0\},y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "# 32  Using M
aple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "dsolve(\{diff(h(x)
,x,x)=6*x,h(0)=3,D(h)(0)=-1\},h(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 71 "#36  If you would rather not type  theta  you can use t (or x, \+
or ...)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "dsolve(\{diff(
y(theta),theta,theta)-y(theta)=exp(theta)-exp(-theta)+2, y(0)=0, D(y)(
0)=0\}, y(theta));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "# 56" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "dsolve(diff(y(x),x,x,x)*2+3*
diff(y(x),x,x)+diff(y(x),x)-4*y(x) =  exp(-x), y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs(\{_C1=0,_C2=0,_C3=0\},\");" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(2*r^3+3*r^2+r-4,r);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 35 "p. 206- 11, 21, 22, 23, 24, 27, 28." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "37 0 0" 0 }
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