{VERSION 5 0 "IBM INTEL NT" "5.0" }
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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "# Solving homogeneo
us linear equations with constant\n# coefficients\n\n# filename = 04.2
.nws\n\nrestart;\nwith(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 84 "\nassume(b^2-4*a*c>0):\nconstcoeffsols(a*diff(y(x),x$
2)+b*diff(y(x),x)+c*y(x)=0,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 105 "a:='a':b:='b':c:='c':\nassume(b^2-4*a*c=0):\nconstco
effsols(a*diff(y(x),x$2)+b*diff(y(x),x)+c*y(x)=0,y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "a:='a':b:='b':c:='c':\nassume(b^2-
4*a*c<0):\nconstcoeffsols(a*diff(y(x),x$2)+b*diff(y(x),x)+c*y(x)=0,y(x
));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 1" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 63 "y:='y':x:='x':\ndiffeqn:=diff(y(x),x$2)+3
*diff(y(x),x)-4*y(x)=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "
with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "constcoef
fsols(diffeqn,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "gen
sol:=dsolve(diffeqn,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
52 "toplot:=[seq(seq(rhs(gensol),_C1=-1..1),_C2=-1..1)];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(toplot,x=-1..1,y=-10..10);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{c1+c2=1,-4*c1+c2=-1
\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "partsol:=dsolve(\{d
iffeqn,y(0)=1,D(y)(0)=-1\},y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "plot(rhs(partsol),x=-1..1);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 9 "Example 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "
gensol:=dsolve(diff(y(x),x$2)+2*diff(y(x),x)+y(x)=0,y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "toplot:=[seq(seq(rhs(gensol),_C1=-1
..1),_C2=-1..1)]:\nplot(toplot,x=-1..1,y=-5..5);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 9 "Example 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 61 "gensol:=dsolve(diff(y(x),x$2)+4*diff(y(x),x)+20*y(x)=0,y(x));" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "partsol:=dsolve(\{diff(y(x
),x$2)+4*diff(y(x),x)+20*y(x)=0,y(0)=3,D(y)(0)=-1\},y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(rhs(partsol),x=-1..1);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 4" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "factor(24*m^3+2*m^2-5*m-1);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 28 "solve(24*m^3+2*m^2-5*m-1=0);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 62 "dea:=24*diff(y(x),x$3)+2*diff(y(x),x$2)-5*di
ff(y(x),x)-y(x)=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "const
coeffsols(dea,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "gen
sola:=dsolve(dea,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "
jvals:=[-1,1]:\ntoplot:=\{seq(seq(subs(\{_C1=j,_C2=1,_C3=i\},rhs(genso
la)),i=-1..1),j=jvals)\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "plot(toplot,x=-2..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "factor(m^3+3*m^2+3*m+1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "solve(m^3+3*m^2+3*m+1=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 85 "deb:=diff(y(x),x$3)+3*diff(y(x),x$2)+3*diff(y(x),x)+y(x)=0:\nc
onstcoeffsols(deb,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 
"gensolb:=dsolve(deb,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
90 "jvals:=[-1,1]:\ntoplot:=\{seq(seq(subs(\{_C1=i,_C2=j,_C3=1\},rhs(g
ensolb)),i=-1..1),j=jvals)\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 29 "plot(toplot,x=-2..3,y=-3..2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "factor(4*m^4+12*m^3+49*m^2+42*m+10,I);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "solve(4*m^4+12*m^3+49*m^2+42*m+10=0
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "dec:=4*diff(y(x),x$4
)+12*diff(y(x),x$3)+49*diff(y(x),x$2)+42*diff(y(x),x)+10*y(x)=0:\ncons
tcoeffsols(dec,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ge
nsolc:=dsolve(dec,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 
"cvals:=[[1,0,1,0],[0,1,0,1],[1,1,0,1],[1,-1,1,2],[0,2,1,-2],[1,-2,1,2
]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "f:=c->subs(\{_C1=c[1
],_C2=c[2],_C3=c[3],_C4=c[4]\},rhs(gensolc)):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "toplot:=map(f,cvals);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 21 "plot(toplot,x=-1..2);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 36 "factor(m^4+4*m^3+24*m^2+40*m+100,I);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "solve(m^4+4*m^3+24*m^2+40*m+100=0,m
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "ded:=diff(y(x),x$4)+
4*diff(y(x),x$3)+24*diff(y(x),x$2)+40*diff(y(x),x)+100*y(x)=0:\nconstc
oeffsols(ded,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "gens
old:=dsolve(ded,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "
cvals:=[[5,0,1,0],[0,1,0,-3],[1,3,0,1],[1,-1,1,2],[0,2,1,-2],[1,-2,5,2
],[0,-3,0,2],[3,0,0,2],[1,1,1,1]]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "f:=c->subs(\{_C1=c[1],_C2=c[2],_C3=c[3],_C4=c[4]\},rh
s(gensold)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "toplot:=map(f,cvals
);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "plot(toplot,x=-1/2..3/2);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 9 "Example 5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "
gensol:=dsolve(4*diff(y(x),x$3)+33*diff(y(x),x)-37*y(x)=0,y(x));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "toplot:=[seq(seq(seq(rhs(gen
sol),_C1=0..1),_C2=-1..1),_C3=-1..0)]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "plot(toplot,x=0..2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 100 "partsol:=dsolve(\{4*diff(y(x),x$3)+33*diff(y(x),x)-3
7*y(x)=0,y(0)=0,D(y)(0)=-1,(D@@2)(y)(0)=3\},y(x));" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 38 "plot(rhs(partsol),x=0..2,y=-0.5..1.5);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 6" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 131 "partsol:=dsolve(\{diff(y(x),x$3)+0.344425*diff(
y(x),x$2)+12.4454*diff(y(x),x)-4.50047*y(x)=0,y(0)=a,D(y)(0)=a,(D@@2)(
y)(0)=b\},y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(r
hs(partsol),5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "constcoe
ffsols(diff(y(x),x$3)+0.344425*diff(y(x),x$2)+12.4454*diff(y(x),x)-4.5
0047*y(x)=0,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "solve
(m^3+0.344425*m^2+12.4454*m-4.50047=0);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 101 "y:=x->c[1]*exp(.3545567566*x)+exp(-.3494908783*x)*(c
[2]*cos(3.545572567*x)+c[3]*sin(-3.545572567*x)):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 39 "sys:=\{y(0)=0,D(y)(0)=a,(D@@2)(y)(0)=b\};" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "cvals:=solve(sys,\{c[1],c[
2],c[3]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sol:=subs(cv
als,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "bval:=solve(.
5349308732e-1*a+.7653001930e-1*b=0,b);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "sol:=subs(b=bval,sol);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "dsol:=diff(sol,x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "plot(dsol/a,x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "simplify(dsol/a);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "critval:=fsolve(simplify(dsol/a=0),x=0.3..0.5);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalf(subs(x=critval,sol));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "avals:=seq(0.5+0.5*i,i=
1..9):\ntoplot:=[seq(subs(a=i,sol),i=avals)];" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "plot(toplot,x=0..6);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 34 "Application:  Testing for Diabetes" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 74 "eq1:=diff(g(t),t)=-a*g(t)-b*h(t)+J(t);\n  eq2:
=diff(h(t),t)=-c*h(t)+d*g(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "step_1:=solve(eq1,h(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "step_2:=diff(step_1,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 52 "step_3:=subs(\{h(t)=step_1,diff(h(t),t)=step_2\},eq2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "step_4:=expand(b*step_3);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "step_5:=subs(\{diff(J(t),t
)=0,J(t)=0\},step_4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "a:
='a':c:='c':b:='b':d:='d':\nsol:=dsolve(step_5,g(t));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "step_6:=subs(\{a+c=2*alpha,a^2-2*c*
a+c^2-4*d*b=-4*omega^2\},rhs(sol));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "step_7:=simplify(step_6,symbolic);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 29 "step_8:=convert(step_7,trig);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "step_9:=convert(step_8,expsincos);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "step_10:=collect(step_9
,\{cos(omega*t),sin(omega*t)\});" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "toapply:=solve(\{_C1+_C2=c[1],-I*_C1+I*_C2=c[2]\},\{_
C1,_C2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "model:=simpli
fy(subs(toapply,step_10));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
4 "?fit" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 \+
0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }