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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 61 "Oby\350ejn\351 diferenci
\341ln\355 rovnice s po\350\355ta\350ovou podporou - Maple" }}{PARA 
19 "" 0 "" {TEXT 271 12 "Petr Kundr\341t" }{TEXT 272 0 "" }}{PARA 256 
"" 0 "" {TEXT -1 32 "\332stav matematiky, FSI VUT v Brn\354" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 262 "Tento soubor vznikl za \372\350elem ilus
trace pou\236it\355 prost\370ed\355 Maple k \370e\232en\355 a vizualiz
aci \370e\232en\355 oby\350ejn\375ch diferenci\341ln\355ch rovnic. Od \+
\350ten\341\370e se p\370edpokl\341d\341 element\341rn\355 orientace v
 prost\370ed\355 Maple a z\341kladn\355 znalosti z oblasti oby\350ejn
\375ch diferenci\341ln\355ch rovnic.   " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "restart; # tento p\370\355kaz vy\350ist\355 pam\354
\235 programu Maple" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "ODR 1. \+
\370\341du" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 298 "V t\351to kapitole se budeme zab
\375vat \370e\232en\355m diferenci\341ln\355ch rovnic prvn\355ho \370
\341du. Uk\341\236eme si, jak formulovat diferenci\341ln\355 rovnici, \+
z\355skat obecn\351 \370e\232en\355, partikul\341rn\355 \370e\232en
\355 po\350\341te\350n\355ho probl\351mu a na z\341v\354r i vykreslen
\355 sm\354rov\351ho pole ODR1. Nejprve tedy naformulujeme oby\350ejno
u diferenci\341ln\355 rovnici   " }{XPPEDIT 18 0 "diff(y(x),x) = -1/2*
y(x);" "6#/-%%diffG6$-%\"yG6#%\"xGF*,$*(\"\"\"F-\"\"#!\"\"-F(6#F*F-F/
" }{TEXT -1 67 "  . Pro efektivn\354j\232\355 pr\341ci s touto rovnic
\355 si ji ulo\236\355me do v\375razu " }{TEXT 256 9 "MojeODR1:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "MojeODR1 := diff(y(x),x) = -
1/2*y(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 10 "Pozn\341mka. " }
{TEXT -1 124 "Je d\371le\236it\351, aby nezn\341m\341 funkce v diferen
ci\341ln\355 rovnici byla zapisov\341na i s jej\355 nez\341visle prom
\354nnou (tedy v na\232em p\370\355kladu " }{XPPEDIT 18 0 "y(x);" "6#-
%\"yG6#%\"xG" }{TEXT -1 155 "). V opa\350n\351m p\370\355pad\354 dojde
 pozd\354ji k chyb\341m, nebo\235  Maple pokl\341d\341 nezn\341mou bez
 nez\341visle prom\354nn\351 za parametr. Chybn\375 z\341pis v\375\232
e formulovan\351 rovnice by byl" }}{PARA 0 "" 0 "" {TEXT 266 35 "> Moj
eODR := diff(y(x),x) = -1/2*y;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Obecn\351 \370e\232en\355 rovnice \+
MojeODR1 z\355sk\341me pou\236it\355m p\370\355kazu " }{TEXT 257 6 "ds
olve" }{TEXT -1 54 ". Parametry tohoto p\370\355kazu jsou diferenci
\341ln\355 rovnice " }{TEXT 261 8 "MojeODR1" }{TEXT -1 56 " a nezn\341
m\341, vzhledem ke kter\351 tuto rovnici \370e\232\355me, tedy " }
{XPPEDIT 18 0 "y(x);" "6#-%\"yG6#%\"xG" }{TEXT -1 33 ". \330e\232en
\355 si lze ulo\236it do v\375razu " }{TEXT 258 12 "ObecneReseni" }
{TEXT -1 54 " prost\370ednictv\355m p\370i\370azen\355 a odkazu na pra
vou stranu (" }{TEXT 259 3 "rhs" }{TEXT -1 25 ") posledn\355ho v\375st
upu (%)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dsolve(MojeODR1
,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ObecneReseni:=rh
s(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Integra\350n\355 konstan
tu v \370e\232en\355 ODR ozna\350uje Maple jako \"" }{TEXT 273 3 "_C1
" }{TEXT -1 106 "\". Nyn\355 se pokus\355me ur\350it partikul\341rn
\355 \370e\232en\355 po\350\341te\350n\355ho probl\351mu, sest\341vaj
\355c\355ho z diferenci\341ln\355 rovnice " }{TEXT 262 8 "MojeODR1" }
{TEXT -1 22 " a po\350\341te\350n\355 podm\355nky " }{XPPEDIT 18 0 "y(
0) = 3;" "6#/-%\"yG6#\"\"!\"\"$" }{TEXT -1 24 ". Pou\236ijeme op\354t \+
p\370\355kaz " }{TEXT 260 6 "dsolve" }{TEXT -1 87 " s parametry: dsolv
e(\{diferenci\341ln\355 rovnice, po\350\341te\350n\355 podm\355nka\}, \+
hledan\341 funkce y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 
"dsolve(\{MojeODR1, y(0)=3\}, y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 131 "Pro vykreslen\355 \370e\232en\355 po\350\341te\350n\355ho prob
l\351mu lze nyn\355 vyu\236\355t jak klasick\375 p\370\355kaz plot pro
 vykreslen\355 prav\351 strany p\370edchoz\355ho v\375razu," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(rhs(%), x= 0 .. 5);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "tak i p\370\355kaz " }{TEXT 263 
6 "DEplot" }{TEXT -1 52 " z knihovny DEtools. Parametry tohoto p\370
\355kazu jsou: " }}{PARA 0 "" 0 "" {TEXT -1 153 "DEplot(diferenci\341l
n\355 rovnice, nezn\341m\341, rozsah grafu v x-ov\351 ose (p\370\355p.
 i y-ov\351 ose), [seznam po\350\341te\350n\355ch podm\355nek], dal
\232\355 voliteln\351 parametry - viz help)" }}{PARA 0 "" 0 "" {TEXT 
-1 76 "Pokud chceme pou\236\355t p\370\355kaz z dosud nezahrnut\351 kn
ihovny (v\375\232e by muselo b\375t \"" }{TEXT 264 14 "with(DEtools);
" }{TEXT -1 38 "\"), m\371\236eme pou\236\355t n\341sleduj\355c\355 sy
ntaxi." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "DEtools[DEplot](
MojeODR1, y(x), x=0..5, [[y(0) = 3],[y(0) = 4]], colour=grey, linecolo
r=[red,blue]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 354 "Jak je z obr
\341zku z\370ejm\351, lze do jednoho takov\351ho grafu vykreslit i v
\355ce partikul\341rn\355ch \370e\232en\355, tedy v na\232em p\370\355
pad\354 y(0) = 3 a y(0) = 4. Nav\355c je v grafu \232edou barvou vykre
sleno pole orientovan\375ch \372se\350ek (sm\354rov\351 pole), kter
\351 maj\355 tu vlastnost, \236e se ve sv\375ch st\370edech dot\375kaj
\355 jednotliv\375ch integr\341ln\355ch k\370ivek p\370\355slu\232ej
\355c\355ch n\354jak\351 po\350\341te\350n\355 podm\355nce.  " }}}
{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT 267 9 "Pozn\341mka." }{TEXT -1 
221 " Hled\341me-li \370e\232en\355 oby\350ejn\351 diferenci\341ln\355
 rovnice, m\371\236eme se programu Maple ot\341zat, o jakou diferenci
\341ln\355 rovnici se jedn\341. K tomu n\341m poslou\236\355 p\370\355
kaz odeadvisor z knihovny DEtools, kter\375 n\341m zadanou rovnici kla
sifikuje. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "DEtools[odead
visor](MojeODR1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "O jednotliv
\375ch  typech rovnic: viz n\341pov\354da k p\370\355kazu odeadvisor. \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "? odeadvisor" }}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 20 "Vyzkou\232ejte si sami:" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 77 "Nalezn\354te partikul\341rn\355 \370e\232
en\355 po\350\341te\350n\355ch probl\351m\371 a nakreslete jejich graf
y." }}{PARA 0 "" 0 "" {TEXT -1 37 "a)   y'-3y = 8x - sin(x),   y(0) = \+
1." }}{PARA 0 "" 0 "" {TEXT -1 5 "b)   " }{XPPEDIT 18 0 "diff(y(x),x) \+
= -4*x*y(x)+2*x*exp(-x^2)*sqrt(y(x));" "6#/-%%diffG6$-%\"yG6#%\"xGF*,&
*(\"\"%\"\"\"F*F.-F(6#F*F.!\"\"**\"\"#F.F*F.-%$expG6#,$*$F*F3F1F.-%%sq
rtG6#-F(6#F*F.F." }{TEXT -1 13 ",   y(0) = 2." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 15 "ODR n-t\351ho \370\341du" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 353 "V t\351to \350\341sti se budeme zab\375vat \370e\232en\355m di
ferenci\341ln\355ch rovnic vy\232\232\355ho \370\341du. Uk\341\236eme \+
si, jak z\355skat obecn\351 \370e\232en\355, partikul\341rn\355 \370e
\232en\355 po\350\341te\350n\355ho probl\351mu a na z\341v\354r i vykr
eslen\355 f\341zov\351ho portr\351tu \370e\232en\355. Jako ilustrativn
\355 p\370\355klad si naformulujeme line\341rn\355 oby\350ejnou difere
nci\341ln\355 rovnici 3. \370\341du s konstantn\355mi koeficienty, kte
rou si ulo\236\355me do v\375razu " }{TEXT 268 9 "MojeODR3:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "MojeODR3 := diff(y(x),x$3)-5
*diff(y(x),x$2)+3*diff(y(x),x) -3*y(x)=0;" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Obecn\351 \370e\232en
\355 rovnice MojeODR z\355sk\341me pou\236it\355m p\370\355kazu " }
{TEXT 269 6 "dsolve" }{TEXT -1 54 ". Parametry tohoto p\370\355kazu js
ou diferenci\341ln\355 rovnice " }{TEXT 270 8 "MojeODR3" }{TEXT -1 55 
" a nezn\341m\341, vzhledem ke kter\351 tuto rovnici \370e\232\355m, t
edy " }{XPPEDIT 18 0 "y(x);" "6#-%\"yG6#%\"xG" }{TEXT -1 2 ". " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dsolve(MojeODR3,y(x));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Pro vy\350\355slen\355 pon\354kud \+
nep\370ehledn\351ho, symbolicky po\350\355tan\351ho v\375razu lze pou
\236\355t p\370\355kaz evalf:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 276 "K vykreslen
\355 grafu \370e\232en\355 MojeODR3 s po\350\341te\350n\355mi podm\355
nkami y(0) = 1, y'(0) = 2, y''(0) = 1 vyu\236ijeme p\370\355kaz DEplot
. Poznamenejme, \236e po\350\341te\350n\355 podm\355nky je t\370eba za
dat ve form\354 funkc\355, a pomoc\355 oper\341toru derivace \"D\". Ne
lze pou\236\355t v\375raz, tedy formu subs(x = 0, diff(y(x), x))." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "DEtools[DEplot](MojeODR3, y
(x), x = 0..2, [[y(0) = 1,D(y)(0) = 2,(D@@2)(y)(0) = 1]], linecolor = \+
[blue],stepsize =.05);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "Vyzkou\232ejte si sami:" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Nalezn\354te obecn\351 \370e\232e
n\355 a \370e\232en\355 po\350\341te\350n\355ch probl\351m\371 n\341sl
eduj\355c\355ch \372loh. Klasifikujte \370e\232en\351 rovnice." }}
{PARA 0 "" 0 "" {TEXT -1 43 "a)  sin(x)y''(x) + 5cos(x)y'(x)-x = 5y(x)
. " }}{PARA 0 "" 0 "" {TEXT -1 66 "b) y'''(x) - tg(x)y(x) = cos(x), y(
0) = 1, y'(0)  = 0, y''(0) = 3." }}{PARA 0 "" 0 "" {TEXT -1 47 "c) y''
(x) - sin(y(x)) = 0, y(1) = 1, y'(1) = 2." }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "Syst\351my ODR 1. \370
\341du" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "V t\351to kapitole budeme \370e
\232it syst\351my oby\350ejn\375ch diferenci\341ln\355ch rovnic 1. \+
\370\341du. Zam\354\370\355me se na syst\351m t\370\355 rovnic 1. \370
\341du:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "MujSystem:=diff(
x(t),t)=x(t)-y(t),diff(y(t),t)=z(t)-y(t),diff(z(t),t)=-3*y(t);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Obecn\351 \370e\232en\355 op\354t \+
z\355sk\341me pomoc\355 p\370\355kazu " }{TEXT 274 6 "dsolve" }{TEXT 
-1 3 ".  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dsolve(\{MujSy
stem\},\{x(t),y(t),z(t)\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "N
\341sleduj\355c\355 grafy n\341m umo\236\362uj\355 zobrazit z\341vislo
sti mezi jednotliv\375mi prom\354nn\375mi t, x(t), y(t), z(t)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "DEplot3d(\{MujSystem\},[x(t),y(t),
z(t)],t=0..10,[[x(0)=0,\ny(0)=1,z(0)=-9]],scene=[t,x(t),y(t)],stepsize
=.1,\ntitle=`Reseni systemu v souradnicich t, x(t), y(t)`,linecolor=t)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "DEplot3d(\{MujSystem
\},[x(t),y(t),z(t)],t=0..10,[[x(0)=0,\ny(0)=1,z(0)=-9]],scene=[t,x(t),
z(t)],stepsize=.1,\ntitle=`Reseni systemu v souradnicich t, x(t), z(t)
`,linecolor=t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "DEplot3
d(\{MujSystem\},[x(t),y(t),z(t)],t=0..10,[[x(0)=0,\ny(0)=1,z(0)=-9]],s
cene=[t,y(t),z(t)],stepsize=.1,\ntitle=`Reseni systemu v souradnicich \+
t, y(t), z(t)`,linecolor=t, axes = normal);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 192 "DEplot3d(\{MujSystem\},[x(t),y(t),z(t)],t=0..10,[
[x(0)=0,\ny(0)=1,z(0)=-9]],scene=[x(t),y(t),z(t)],stepsize=.1,\ntitle=
`Reseni systemu v souradnicich x(t), y(t), z(t)`,linecolor=t, axes = n
ormal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 188 "Pro vykreslen\355 dvo
urozm\354rn\351ho grafu (ortogon\341ln\355 projekce \370e\232en\355 do
 roviny dvou \370e\232en\375ch prom\354nn\375ch) lze vyu\236\355t p
\370\355kazu phaseportrait. N\355\236e vid\355me \370e\232en\355 soust
avy MujSystem v rovin\354 (y,z):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 123 "phaseportrait([MujSystem],[x(t),y(t),z(t)],t=-2.5..9
,[[x(0)=0,y(0)=1,z(0)=-9]],stepsize=.05,scene=[y(t),z(t)],linecolor=t)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 48 "Stejnou situaci uvid\355te vhodn\375m nato\350en\355
m grafu " }{TEXT 275 8 "DEplot3d" }{TEXT -1 38 " v sou\370adnic\355ch \+
y(t), z(t) (viz v\375\232e)." }}}}}{MARK "8" 0 }{VIEWOPTS 1 1 0 3 2 
1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }