Caderno de equuações diferenciais

CADERNO DE EXERCÍCIOS 2 OPTIMIZAÇÃO

EQUAÇÕES DIFERENCIAIS ORDINÁRIAS

Licenciaturas em Gestão, Gestão e Engenharia Industrial, Marketing e Finanças

ISCTE-IUL

MARÇO 2010

4
414

Htxdêøhv glihuhqfldlv ruglqäuldv +HGR,
H{hufðflrv sursrvwrv
Jhqhudolgdghv

41414

41 Xpd HGR fruuhvsrqgh d xpd uhodêær U {> |> |  > |  > | > = = = > | Eq @ 3 hqwuh d yduläyho lqghshqghqwh {/ d ixqêær | @ |+{, +ghvljqdgd sru yduläyho ghshqghqwh, h dojxpdv gdv vxdv ghulydgdv hp rughp d {1 Uhvroyhu xpd HGR ì ghwhuplqdu +sru sulplwlydêær, xpd idpðold gh ixqêøhv | txh d yhultxh/ ghvljqdgd sru lqwhjudo jhudo rx vroxêær jhudo gd HGR1 Frqvlghuh d HGR |Eq frv { @ 3=

+e, Jhrphwulfdphqwh/ r lqwhjudo jhudo uhsuhvhqwd xpd idpðold gh fxu0 ydv/ ghqrplqdgdv fxuydv lqwhjudlv gd HGR1 D fdgd ydoru gh F 5 U qr lqwhjudo jhudo fruuhvsrqgh xpd fxuyd lqwhjudo1 Uhsuhvhqwh dojxpdv gdv fxuydv lqwhjudlv gd htxdêær | frv +{, @ 3=

+d, Uhvroyd d htxdêær sdud q @ 4= ^Vroxêær= r lqwhjudo jhudo ì | @ vlq +{, . F/ frp F 5 U`

^Vroxêøhv= sru h{hpsor/ d fxuyd lqwhjudo sdud F @ 3 ì r juäfr gd ixqêær | @ vlq +{,/ d fxuyd lqwhjudo sdud F @ 4 ì r juäfr gd ixqêær | @ vlq +{, 4/ d fxuyd lqwhjudo sdud F @ 3=8 ì r juäfr gd ixqêær | @ 3=8 . vlq +{,/ hwf1` +f, Prvwuh txh | @ frv+{, . F{ . F  / frp F> F  5 U ì r lqwhjudo jhudo gd HGR txdqgr q @ 5= +g, Ghwhuplqh r lqwhjudo jhudo gd htxdêær |  frv +{, @ 31 ^Vroxêær= r lqwhjudo jhudo ì | @ vlq +{, . F{2 @5 . F  { . F  / frp F> F  > F  5 U`

+h, Vhpsuh txh ì srvvðyho ghwhuplqdu r ydoru gdv frqvwdqwhv F 5 U/ rewhprv xp grv lqwhjudlv sduwlfxoduhv rx vroxêøhv sduwlfxoduhv gd HGR1 Fdofxoh r lqwhjudo sduwlfxodu gd HGR |  frv +{, @ 3 vxmhlwd ãv frqglêøhv +v1f1, |  +, @ 5 a | +, @ 4 4 a
{ { 4 6{| . 8|  . |  @ 5 {   g| @ 5 . vlq +{, = 6 +{ . 4, | e . 8 g{

h

^Vroxêøhv= rughp 6 h judx 5/ rughp 5 h judx 4/ rughp 4 h judx 6` 61 Xpd HGR gh 4d rughp U +{> |> | , @ 3 gl}0vh qd irupd gh Sid txdqgr ì hvfulwd frpr S +{> |,g{ . T+{>