1.1 DEFINICIONES Y TERMINOLOGÍA
27. y 2y 0
28. 5y 2y
29. y 5y 6y 0
30. 2y 7y 4y 0
(Q ORV SUREOHPDV \ GHWHUPLQH ORV YDORUHV GH m SDUD TXH OD IXQFLyQ y xm VHD XQD VROXFLyQ GH OD HFXDFLyQ GLIHUHQFLDO GDGD 31. xy 2y 0 32. x2y 7xy 15y 0 (Q ORV SUREOHPDV GHO DO HPSOHH HO FRQFHSWR GH TXH y c, x HV XQD IXQFLyQ FRQVWDQWH VL \ VyOR VL y SDUD GHWHUPLQDU VL OD HFXDFLyQ GLIHUHQFLDO GDGD WLHQH VROXFLRQHV FRQVWDQWHV 33. 3xy 5y 10 34. y y 2 2y 3 35. (y 1)y 1 36. y 4y 6y 10 (Q ORV SUREOHPDV \ FRPSUXHEH TXH HO SDU GH IXQFLRQHV TXH VH LQGLFD HV XQD VROXFLyQ GHO VLVWHPD GDGR GH HFXDFLRQHV GLIHUHQFLDOHV HQ HO LQWHUYDOR , ). 37. dx dt
x
dy 5x dt x e 2t y
e
2 38. d x dt 2
3y 3y; 3e6t, 2t
5e6t
4y
d 2y 4x dt 2 x cos 2t y
cos 2t
l
11
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y
et 1
et; sen 2 t sen 2 t
1 5
1 5
et
Problemas para analizar
FIGURA 1.1.6 *Ui¿FD SDUD HO SUREOHPD
y
46.
39. &RQVWUX\D XQD HFXDFLyQ GLIHUHQFLDO TXH QR WHQJD DOJXQD VROXFLyQ UHDO
1
40. &RQVWUX\D XQD HFXDFLyQ GLIHUHQFLDO TXH HVWp VHJXUR TXH VRODPHQWH WLHQH OD VROXFLyQ WULYLDO y ([SOLTXH VX UD]RQDPLHQWR 41. ¢4Xp IXQFLyQ FRQRFH GH FiOFXOR FX\D SULPHUD GHULYDGD VHD HOOD PLVPD" ¢6X SULPHUD GHULYDGD HV XQ P~OWLSOR FRQVWDQWH k GH Vt PLVPD" (VFULED FDGD UHVSXHVWD HQ IRUPD GH XQD HFXDFLyQ GLIHUHQFLDO GH SULPHU RUGHQ FRQ XQD VROXFLyQ 42. ¢4Xp IXQFLyQ R IXQFLRQHV GH FiOFXOR FRQRFH FX\D VHJXQGD GHULYDGD VHD HOOD PLVPD" ¢6X VHJXQGD GHULYDGD HV OD QHJDWLYD GH Vt PLVPD" (VFULED FDGD UHVSXHVWD HQ IRUPD GH XQD HFXDFLyQ GLIHUHQFLDO GH VHJXQGR RUGHQ FRQ XQD VROXFLyQ
x
1
et,
1
x
FIGURA 1.1.7 *Ui¿FD SDUD HO SUREOHPD 47. /DV JUi¿FDV GH ORV PLHPEURV GH XQD IDPLOLD XQLSDUDPpWULFD x3 y3 3cxy VH OODPDQ folium de Descartes. &RPSUXHEH TXH HVWD IDPLOLD HV XQD VROXFLyQ LPSOtFLWD GH OD HFXDFLyQ GLIHUHQFLDO GH SULPHU RUGHQ
dy dx
y(y3 2x3) x(2y3 x3)