{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Tit le" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 43 "K\344yr\344parven differe ntiaaliyht\344l\366n etsiminen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "Olkoon annettuna k\344yr\344parvi \n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "kayraparvi:= (x+_C1)^2+(y+_C 2)^2=_C3^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 7 "miss\344 _" }{TEXT 260 2 "C1" }{TEXT -1 3 ", _" }{TEXT 259 2 "C2" }{TEXT -1 5 " ja _" }{TEXT 258 2 "C3" }{TEXT -1 181 " ovat \+ parven parametrit. Jokaisella (ainakin l\344hes) n\344iden arvokombina atiolla parvesta saatava yht\344l\366 esitt\344\344 tason ympyr\344 \344. Toisaalta tason jokaisen ympyr\344n yht\344l\366 on t\344t\344 m uotoa. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Tulkitaan " }{TEXT 256 1 "y" }{TEXT -1 24 " funktioksi muuttujasta " }{TEXT 257 1 "x" }{TEXT -1 58 " tavoitteena l\366yt\344\344 differe ntiaaliyht\344l\366 t\344lle funktiolle:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "parvi:= subs(y=y(x), kayraparvi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Derivoidaan t\344m\344 yht\344l\366 kolmesti, mink \344 j\344lkeen saaduista nelj\344st\344 yht\344l\366st\344 eliminoida an parametrit _" }{TEXT 262 2 "C1" }{TEXT -1 3 ", _" }{TEXT 263 2 "C2 " }{TEXT -1 5 " ja _" }{TEXT 261 2 "C3" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "parvi1 :=diff(parvi, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "parvi2 :=diff(parvi, x$2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "parv i3:=diff(parvi, x$3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "ra tk:= eliminate(\{parvi, parvi1, parvi2, parvi3\}, \{_C1, _C2, _C3\}); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 9 "Eliminate" }{TEXT -1 156 "-komento antaa listan, jonka viimeisen \344 alkiona on eliminoinnin tulos polynomin muodossa. Ratkaisu muunne taan yht\344l\366muotoon merkitsem\344ll\344 polynomi nollaksi." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "diffyht:= op(ratk[-1])=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "Tuloksena on kolmannen kertaluvun d ifferentiaaliyht\344l\366, jonka yleisen\344 ratkaisuna on alkuper\344 inen k\344yr\344parvi. Kyseess\344 on siten kaikkien tason ympyr\366id en differentiaaliyht\344l\366." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 66 "Saadulla differentiaaliyht\344l\366ll\344 on kuitenkin muitakin ratkaisuja:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "simplify(subs(y(x)=_D1*x+_D 2, diffyht));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 19 "SKK & MS 31.05. 2001" }{TEXT -1 1 " " }}}}{MARK "0 2 0" 18 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }