{VERSION 5 0 "IBM INTEL NT" "5.0" }
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1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 
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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "# Method of undeter
mined coefficients to get a\n# particular solution to non-homogeneous \+
linear\n# equations\n\n# filename = 04.4.mws\n\nrestart;\n\n?DEtools" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 1" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 64 "Hom_Sol:=dsolve(diff(y(x),x$2)+5*diff(y(x),x)+
6*y(x)=0,\n\011\011y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "yp:=x->A*exp(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "LHS
eqn:=diff(yp(x),x$2)+5*diff(yp(x),x)+6*yp(x);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 43 "match(2*exp(x)=12*A*exp(x),x,'Val'):\n  Val;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "assign(Val):\n  yp(x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Gen_Sol:=dsolve(diff(y(x),x$
2)+5*diff(y(x),x)+\n\011\0116*y(x)=2*exp(x),y(x));" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 89 "to_plot:=[seq(seq(rhs(Gen_Sol),\n\011_C1=-1
..1),_C2=-1..1)]:\n  plot(to_plot,x=-3..5,y=-3..5);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 40 "A:='A':yp:='yp':\n  yp:=x->A*x*exp(-2*x):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Diff(yp(x),x)=diff(yp(x
),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff(yp(x),x$2)=di
ff(yp(x),x$2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "eqn:=diff
(yp(x),x$2)+5*diff(yp(x),x)+6*yp(x)=3*exp(-2*x);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 19 "aval:=solve(eqn,A);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 103 "y:='y':\n  part_sol:=dsolve(\{diff(y(x),x$2)+5*
diff(y(x),x)+\n\0116*y(x)=3*exp(-2*x),y(0)=a,D(y)(0)=b\},y(x));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "to_plot:=[seq(seq(rhs(part_s
ol),\n\011a=0..2),b=-1..1)]:\n  plot(to_plot,x=-1/2..3/2);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 9 "Example 2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "y:='y':\n  Hom_sol:=dsolve(diff(y(x),x$3)+4*diff(y(x)
,x$2)+\n\01114*diff(y(x),x)+20*y(x)=0,y(x));" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 79 "A:='A':B:='B':C:='C':\n  yp:=x->A*x*exp(-2*x)+x*
exp(-x)*(B*cos(3*x)+C*sin(3*x)):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 110 "eqn:=simplify(diff(yp(x),x$3)+4*diff(yp(x),x$2)+\n
\01114*diff(yp(x),x)+20*yp(x))=10*exp(-2*x)-\n\011\011exp(-x)*cos(3*x)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn1:=eval(subs(x=0,eq
n));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn2:=eval(subs(x=P
i/6,eqn));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn3:=eval(su
bs(x=Pi/3,eqn));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "consts:
=solve(\{eqn1,eqn2,eqn3\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "Part_sol:=subs(consts,yp(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "y:=rhs(Hom_sol)+Part_sol;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "eq1:=eval(subs(x=0,y))=5;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "eq2:=eval(subs(x=0,diff(y,x)))=0;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eq3:=eval(subs(x=0,diff(y,x$2)))=-1
/2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "cvals:=solve(\{eq1,e
q2,eq3\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "assign(cvals)
:\ny;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot(y,x=0..2);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "y:='y':\n  Gen_Sol:=dsolv
e(diff(y(x),x$3)+4*diff(y(x),x$2)+\n\01114*diff(y(x),x)+20*y(x)=\n\011
10*exp(-2*x)-exp(-x)*cos(3*x),y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "combine(Gen_Sol,trig);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 191 "y:='y':\n  Init_Sol:=dsolve(\{diff(y(x),x$3)+4*diff(
y(x),x$2)+\n\01114*diff(y(x),x)+20*y(x)=10*exp(-2*x)-exp(-x)*cos(3*x),
\n\011y(0)=y0,D(y)(0)=y1,(D@@2)(y)(0)=y2\},y(x)):\nstep2:=combine(Init
_Sol,trig);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "assign(Init
_Sol):\n  y0_vals:=\{-1,2\}:y1_vals:=\{-3,0,3\}:y2_vals:=\{-2,2\}:\n  \+
to_plot:=\{seq(seq(seq(rhs(step2),\n\011y0=y0_vals),y1=y1_vals),y2=y2_
vals)\}:\n  plot(to_plot,x=-1/2..3/2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 9 "Example 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "y:
='y':\nhomsol:=dsolve(4*diff(y(x),x$2)+4*diff(y(x),x)+37*y(x)=0,y(x));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A:='A':B:='B':\nyp:=x->
A*cos(3*x)+B*sin(3*x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "s
tep1:=simplify(4*diff(yp(x),x$2)+4*diff(yp(x),x)+37*yp(x))=cos(3*x);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eq1:=eval(subs(x=0,step1)
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eq2:=eval(subs(x=Pi/6
,step1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "vals:=solve(\{
eq1,eq2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "assign(vals)
:\nyp(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "y:='y':\ny:=rh
s(homsol)+yp(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "eval(su
bs(x=0,y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eval(subs(x=
Pi,y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "cval:=solve(eval
(subs(x=0,y))=eval(subs(x=Pi,y)));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "sol:=subs(_C4=cval,y);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "toplot:=\{seq(sol,_C5=-5..5)\}:\nplot(toplot,x=0..2*P
i);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 4" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 53 "x:='x':\nhomsol:=dsolve(diff(x(t),t$2)+4*
x(t)=0,x(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A:='A':B:=
'B':\nxp:=t->A*cos(omega*t)+B*sin(omega*t):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 54 "step1:=simplify(diff(xp(t),t$2)+4*xp(t))=sin(omega
*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn1:=eval(subs(t=0
,step1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eqn2:=eval(sub
s(t=Pi/(2*omega),step1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "vals:=solve(\{eqn1,eqn2\},\{A,B\});" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "x:='x':\nassign(vals):\nx:=rhs(homsol)+xp(t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "cvals:=solve(\{eval(subs(t=0
,x))=1,eval(subs(t=0,diff(x,t)))=0\},\{_C4,_C5\});" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 20 "sola:=subs(cvals,x);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 99 "A:='A':B:='B':\nxp:=t->t*(A*cos(2*t)+B*sin(2*t
)):\nstep1:=simplify(diff(xp(t),t$2)+4*xp(t))=sin(2*t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn1:=eval(subs(t=0,step1));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn2:=eval(subs(t=Pi/12,step
1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "vals:=solve(\{eqn1,
eqn2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "assign(vals):\n
x:=rhs(homsol)+xp(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "cv
als:=solve(\{eval(subs(t=0,x))=1,eval(subs(t=0,diff(x,t)))=0\},\{_C4,_
C5\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solb:=subs(cvals,
x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "x:='x':t:='t':\nsolc
:=dsolve(\{diff(x(t),t$2)+4*x(t)=sin(omega*t),x(0)=1,D(x)(0)=0\},x(t))
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "omegavals:=[seq(.9+.25
*i,i=0..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "somegraphs:
=[seq(plot(subs(omega=i,sola),t=0..18*Pi,x=-1..1,\n\011\011color=BLACK
),i=omegavals)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "with(pl
ots):\n  anarray:=display(somegraphs,insequence=true):\n  display(anar
ray);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(solb,t=0..18*
Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "sold:=dsolve(\{diff
(x(t),t$2)+4*x(t)=sin(2*t),x(0)=1,D(x)(0)=0\},x(t));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "combine(sold,trig);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "73 0 0" 19 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }