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l
CAPÍTULO 1
INTRODUCCIÓN A LAS ECUACIONES DIFERENCIALES
y
(Q WRGRV ORV HMHPSORV DQWHULRUHV KHPRV XVDGR x \ y SDUD GHQRWDU ODV YDULDEOHV LQGHSHQGLHQWH \ GHSHQGLHQWH UHVSHFWLYDPHQWH 3HUR GHEHUtD DFRVWXPEUDUVH D YHU \ WUDEDMDU FRQ RWURV VtPERORV TXH GHQRWDQ HVWDV YDULDEOHV 3RU HMHPSOR SRGUtDPRV GHQRWDU OD YDULDEOH LQGHSHQGLHQWH SRU t \ OD YDULDEOH GHSHQGLHQWH SRU x. x
FIGURA 1.1.4
$OJXQDV VROXFLRQHV GH OD (' GHO LQFLVR E GHO HMHPSOR
EJEMPLO 7
Usando diferentes símbolos
/DV IXQFLRQHV x c1 FRV t \ x c2 VHQ t GRQGH c1 \ c2 VRQ FRQVWDQWHV DUELWUDULDV R SDUiPHWURV VRQ DPEDV VROXFLRQHV GH OD HFXDFLyQ GLIHUHQFLDO OLQHDO x
16x
0.
3DUD x c1 FRV t ODV GRV SULPHUDV GHULYDGDV UHVSHFWR D t VRQ x 4c1 VHQ t \ x 16c1 FRV t. 6XVWLWX\HQGR HQWRQFHV D x \ x VH REWLHQH x
16x
16c1 cos 4t
16(c1 cos 4t)
0.
'H PDQHUD SDUHFLGD SDUD x c2 VHQ t WHQHPRV x 16c 2 VHQ t \ DVt x
16x
16c2 sen 4t
16(c2 sen 4t)
0.
)LQDOPHQWH HV VHQFLOOR FRPSUREDU GLUHFWDPHQWH TXH OD FRPELQDFLyQ OLQHDO GH VROXFLRQHV R OD IDPLOLD GH GRV SDUiPHWURV x c1 FRV t c2 VHQ t HV WDPELpQ XQD VROXFLyQ GH OD HFXDFLyQ GLIHUHQFLDO (O VLJXLHQWH HMHPSOR PXHVWUD TXH OD VROXFLyQ GH XQD HFXDFLyQ GLIHUHQFLDO SXHGH VHU XQD IXQFLyQ GH¿QLGD SRU SDUWHV
EJEMPLO 8
8QD VROXFLyQ GH¿QLGD SRU SDUWHV
/D IDPLOLD XQLSDUDPpWULFD GH IXQFLRQHV PRQRPLDOHV FXiUWLFDV y cx4 HV XQD VROXFLyQ H[SOtFLWD GH OD HFXDFLyQ OLQHDO GH SULPHU RUGHQ xy 4y 0 HQ HO LQWHUYDOR , &RPSUXHEH /DV FXUYDV VROXFLyQ D]XO \ URMD TXH VH PXHVWUDQ HQ OD ¿JXUD D VRQ ODV JUi¿FDV GH y = x4 \ y = x4 \ FRUUHVSRQGHQ D ODV HOHFFLRQHV GH c \ c = UHVSHFWLYDPHQWH
y
/D IXQFLyQ GHULYDEOH GH¿QLGD SRU WUDPRV c 1
x4, x4,
y x c 1
x x
0 0
HV WDPELpQ XQD VROXFLyQ SDUWLFXODU GH OD HFXDFLyQ SHUR QR VH SXHGH REWHQHU GH OD IDPLOLD y cx4 SRU XQD VROD HOHFFLyQ GH c OD VROXFLyQ VH FRQVWUX\H D SDUWLU GH OD IDPLOLD HOLJLHQGR c SDUD x \ c SDUD x 9HD OD ¿JXUD E
a) dos soluciones explicitas y c 1, x d0 x c 1, x 0
b) solución definida en partes
FIGURA 1.1.5
$OJXQDV VROXFLRQHV GH OD (' GHO HMHPSOR
SISTEMAS DE ECUACIONES DIFERENCIALES +DVWD HVWH PRPHQWR KHPRV DQDOL]DGR VyOR HFXDFLRQHV GLIHUHQFLDOHV TXH FRQWLHQHQ XQD IXQFLyQ LQFyJQLWD 3HUR FRQ IUHFXHQFLD HQ OD WHRUtD DVt FRPR HQ PXFKDV DSOLFDFLRQHV GHEHPRV WUDWDU FRQ VLVWHPDV GH HFXDFLRQHV GLIHUHQFLDOHV 8Q sistema de ecuaciones diferenciales ordinarias WLHQH GRV R PiV HFXDFLRQHV TXH LPSOLFDQ GHULYDGDV GH GRV R PiV IXQFLRQHV LQFyJQLWDV GH XQD VROD YDULDEOH LQGHSHQGLHQWH 3RU HMHPSOR VL x \ y GHQRWDQ D ODV YDULDEOHV GHSHQGLHQWHV \ t GHQRWD D OD YDULDEOH LQGHSHQGLHQWH HQWRQFHV XQ VLVWHPD GH GRV HFXDFLRQHV GLIHUHQFLDOHV GH SULPHU RUGHQ HVWi GDGR SRU
dx dt
f(t, x, y)
dy dt
g(t, x, y).
(9)