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{COM_R 2 0{TEXT 1 51 "CHAPTER 3 HIGHLIGHTS\015Dr. Denise Kirschne
r, MATH 308"}}{INP_R 3 0 "> "{TEXT 0 26 "with(DEtools):with(plots
):"}}{INP_R 4 0 "> "{TEXT 0 47 "diffeq:=diff(y(x),x,x)+2*diff(y(x
),x)-3*y(x)=0;"}}{OUT_R 5 0 4{DAG :3n4\`diffeq`=3+7(3n4\`diff`,3(
3p7,3(3n3\`y`,2n3\`x`p12p12j2x0001pBj2x0002pEi2x0003j2x0000}}
{INP_R 6 0 "> "{TEXT 0 20 "Char_eq:=r^2+2*r-3; "}}{OUT_R 7 0 6
{DAG :3n4\`Char_eq`+7*3n3\`r`j2x0002j2x0001p6p8i2x0003pA}}{INP_R 
8 0 "> "{TEXT 0 15 "ev:=solve(\",r);"}}{OUT_R 9 0 8{DAG :3n3\`ev`
,3j2x0001i2x0003}}{INP_R 10 0 "> "{TEXT 0 37 "y(x)=C1*exp(ev[1]*x
)+C2*exp(ev[2]*x);"}}{OUT_R 11 0 10{DAG =3(3n3\`y`,2n3\`x`+5*5n3\
`C1`j2x0001(3n3\`exp`p4pBpB*5n3\`C2`pB(3pE,2+3p5i2x0003pBpB}}
{INP_R 12 0 "> "{TEXT 0 25 "subs(\",diffeq):expand(\");"}}{OUT_R 
13 0 12{DAG =3j2x0000p1}}{INP_R 14 0 "> "{TEXT 0 44 "plot(1*exp(e
v[1]*x)+2*exp(ev[2]*x), x=0..5);"}}{INP_R 15 0 "> "{TEXT 0 72 "di
ffeq:=diff(y(x),x$2)+4*diff(y(x),x)+3*y(x)=0;inits:=y(0)=2,D(y)(0
)=-1;"}}{OUT_R 16 0 15{DAG :3n4\`diffeq`=3+7(3n4\`diff`,3(3p7,3(3
n3\`y`,2n3\`x`p12p12j2x0001pBj2x0004pEj2x0003j2x0000}}{OUT_R 17 0
 15{DAG :3n4\`inits`,3=3(3n3\`y`,2j2x0000j2x0002=3(3(3n3\`D`,2p7p
9i2x0001}}{INP_R 18 0 "> "{TEXT 0 33 "sol:=dsolve(\{diffeq,inits\
},y(x));"}}{OUT_R 19 0 18{DAG :3n3\`sol`=3(3n3\`y`,2n3\`x`+5(3n3\
`exp`,2+3p8i2x0003/3i2x0001j2x0002(3pC,2+3p8p14/3j2x0005p16}}
{INP_R 20 0 "> "{TEXT 0 25 "subs(\",diffeq):expand(\");"}}{OUT_R 
21 0 20{DAG =3j2x0000p1}}{INP_R 22 0 "> "{TEXT 0 24 "plot(rhs(sol
), x=0..10);"}}{COM_R 23 0{TEXT 1 67 "NEXT, we will graph the sol
ution to the one we solved in class:\015\015\015\015"}}{INP_R 24 
0 "> "{TEXT 0 73 "diffeq2:=diff(y(x),x$2)-2*diff(y(x),x)+6*y(x)=0
;inits2:=y(0)=1,D(y)(0)=3;"}}{OUT_R 25 0 24{DAG :3n4\`diffeq2`=3+
7(3n4\`diff`,3(3p7,3(3n3\`y`,2n3\`x`p12p12j2x0001pBi2x0002pEj2x00
06j2x0000}}{OUT_R 26 0 24{DAG :3n4\`inits2`,3=3(3n3\`y`,2j2x0000j
2x0001=3(3(3n3\`D`,2p7p9j2x0003}}{INP_R 27 0 "> "{TEXT 0 38 "sol2
:=dsolve(\{diffeq2, inits2\}, y(x));"}}{OUT_R 28 0 27{DAG :3n4\`s
ol2`=3(3n3\`y`,2n3\`x`+5*5(3n3\`exp`p8j2x0001(3n3\`cos`,2*5j2x000
5/3p11j2x0002p9p11p11p11*7p18p1ApDp11(3n3\`sin`p16p11/3p1Cp18}}
{INP_R 29 0 "> "{TEXT 0 25 "plot(rhs(sol2), x=0..10);"}}{COM_R 30
 0{TEXT 1 67 "NOW, WE  CAN LOOK at the WRONSKIAN: \015WHAT DO WE \+
NEED TO DO THIS???\015"}}{INP_R 31 0 "> "{TEXT 0 13 "with(linalg)
:"}}{OUT_R 32 0 31{TEXT 2 71 "Warning: new definition for   norm\
012Warning: new definition for   trace\012"}}{INP_R 33 0 "> "
{TEXT 0 38 "Wronskian([exp(-2*x),x*exp(-2*x)],x); "}}{OUT_R 34 0 
33{DAG (3n4\`MATRIX`,2[2,3[2,3(3n3\`exp`,2+3n3\`x`i2x0002*5pEj2x0
001p9p14[2,3+3p9p10+5p9p14p12p10}}{INP_R 35 0 "> "{TEXT 0 7 "det(
\");"}}{OUT_R 36 0 35{DAG *3(3n3\`exp`,2+3n3\`x`i2x0002j2x0002}}
{COM_R 37 0{TEXT 1 50 "REDUCTION OF ORDER IS NEXT, we need one so
lution!\015"}}{INP_R 38 0 "> "{TEXT 0 54 "diffeq:=x^2*diff(y(x),x
$2)+2*x*diff(y(x),x)-2*y(x)=0; "}}{OUT_R 39 0 38{DAG :3n4\`diffeq
`=3+7*5n3\`x`j2x0002(3n4\`diff`,3(3pC,3(3n3\`y`,2p7p7p7j2x0001p1A
*5p7p1Ap10p1Ap9p13i2x0002j2x0000}}{INP_R 40 0 "> "{TEXT 0 13 "sol
1:=y(x)=x;"}}{OUT_R 41 0 40{DAG :3n4\`sol1`=3(3n3\`y`,2n3\`x`p9}}
{INP_R 42 0 "> "{TEXT 0 29 "subs(sol1,diffeq); expand(\");"}}
{OUT_R 43 0 42{DAG =3+7*5n3\`x`j2x0002(3n4\`diff`,3(3p8,3p3p3p3j2
x0001p12*5p3p12pCp12p5p3i2x0002j2x0000}}{OUT_R 44 0 42{DAG =3j2x0
000p1}}{INP_R 45 0 "> "{TEXT 0 33 "subs(y(x)=v(x)*rhs(sol1),diffe
q);"}}{OUT_R 46 0 45{DAG =3+7*5n3\`x`j2x0002(3n4\`diff`,3(3p8,3*5
(3n3\`v`,2p3j2x0001p3p15p3p3p15p15*5p3p15pCp15p5pFi2x0002j2x0000}
}{INP_R 47 0 "> "{TEXT 0 10 "expand(\");"}}{OUT_R 48 0 47{DAG =3+
5*5n3\`x`j2x0003(3n4\`diff`,3(3p8,3(3n3\`v`,2p3p3p3j2x0001p16*5p3
j2x0002pCp16j2x0004j2x0000}}{INP_R 49 0 "> "{TEXT 0 35 "reduced:=
subs(diff(v(x),x)=z(x),\");"}}{OUT_R 50 0 49{DAG :3n4\`reduced`=3
+5*5n3\`x`j2x0003(3n4\`diff`,3(3n3\`z`,2p7p7j2x0001p16*5p7j2x0002
p10p16j2x0004j2x0000}}{COM_R 51 0{TEXT 1 35 "REMEBER! THIS IS ALW
AYS SEPERABLE!!"}}{INP_R 52 0 "> "{TEXT 0 21 "dsolve(reduced,z(x)
);"}}{OUT_R 53 0 52{DAG =3(3n3\`z`,2n3\`x`*5p5i2x0004n3\`_C1`j2x0
001}}{INP_R 54 0 "> "{TEXT 0 22 "v(x)=int(rhs(\"),x)+C2;"}}{COM_R
 55 0{TEXT 1 87 "IT MAY BE HELPFUL TO USE THE SIMPLIFY COMMAND HE
RE such as simplify(int(rhs(....etc..))"}}{OUT_R 56 0 54{DAG =3(3
n3\`v`,2n3\`x`+5*5p5i2x0003n3\`_C1`j2x0001/3i2x0001j2x0003n3\`C2`
pE}}{INP_R 57 0 "> "{TEXT 0 39 "rhs(sol1)*rhs(\"); sol2:=y(x)=exp
and(\");"}}{OUT_R 58 0 57{DAG *5n3\`x`j2x0001+5*5p1i2x0003n3\`_C1
`p3/3i2x0001j2x0003n3\`C2`p3p3}}{OUT_R 59 0 57{DAG :3n4\`sol2`=3(
3n3\`y`,2n3\`x`+5*5p9i2x0002n3\`_C1`j2x0001/3i2x0001j2x0003*5p9p1
2n3\`C2`p12p12}}{INP_R 60 0 "> "{TEXT 0 29 "subs(sol2,diffeq): ex
pand(\");"}}{OUT_R 61 0 60{DAG =3j2x0000p1}}{COM_R 62 0{TEXT 1 38
 "WHAT IS THE LONGTERM BEHAVIOR OF y????"}}{INP_R 63 0 "> "{TEXT 
0 44 "_C1:=1; C2:=2;sol2;plot(rhs(sol2), x=1..10);"}}{OUT_R 64 0 
63{DAG :3n3\`_C1`j2x0001}}{OUT_R 65 0 63{DAG :3n3\`C2`j2x0002}}
{OUT_R 66 0 63{DAG =3(3n3\`y`,2n3\`x`+5*3p5i2x0002/3i2x0001j2x000
3p5j2x0002}}{INP_R 67 0 "> "{TEXT 0 0 ""}}{INP_R 68 0 "> "{TEXT 0
 0 ""}}}{END}