05:35, 1 ÑÐ°Ð½ÑÐ°ÑÐ¸ 1970

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$Fractional and operational calculus with generalized fractional ...$

$Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...$

68 Boro Piperevski 2ϕ 2 Z ′′′ 2 +3ϕ[ϕ′ +8(x 2 + e 2 x + e 1 e 3 )]Z ′′ 2 + {ϕ′′ ϕ + 16(2x + e 2 )ϕ+ +8(x 2 + e 2 x + e 1 e 3 )ϕ ′ + 64(x 2 + e 2 x + e 1 e 3 ) 2 − 8[n(n +1)x − λ]ϕ}Z ′ 2+ + {8ϕ − 32[n(n +1)− λ](x 2 + e 2 x + e 1 e 3 )+ + 32(2x + e 2 )(x 2 + e 2 x + e 1 e 3 ) − 4n(n +1)ϕ}Z 2 =0, (II 1 ) 2ϕ 2 Z ′′′ 3 +3ϕ[ϕ ′ +8(x 2 + e 3 x + e 1 e 2 )]Z ′′ 3 + {ϕ ′′ ϕ + 16(2x + e 3 )ϕ+ +8(x 2 + e 3 x + e 1 e 2 )ϕ ′ + 64(x 2 + e 3 x + e 1 e 2 ) 2 − 8[n(n +1)x − λ]ϕ}Z ′ 3+ + {8ϕ − 32[n(n +1)− λ](x 2 + e 3 x + e 1 e 2 )+ + 32(2x + e 3 )(x 2 + e 3 x + e 1 e 2 ) − 4n(n +1)ϕ}Z 3 =0, (III 1 ) 2ϕ 2 Z ′′′ 4 +3ϕ[ϕ ′ +8(2x 2 − e 3 x − e 2 3)]Z ′′ 4 + {ϕ ′′ ϕ + 16(5x − 2e 3 )ϕ+ +8(2x 2 − e 3 x − e 2 3 )ϕ′ + 64(2x 2 − e 3 x − e 2 3 )2 − 8[n(n +1)x − λ]ϕ}Z ′ 4 + + {24ϕ − 32[n(n +1)− λ](2x 2 − e 3 x − e 2 3)+ + 32(6x +3e 3 )(2x 2 − e 3 x − e 2 3) − 4n(n +1)ϕ}Z 4 =0, (IV 1 ) 2ϕ 2 Z ′′′ 5 +3ϕ[ϕ ′ +8(2x 2 − e 2 x − e 2 2)]Z ′′ 5 + {ϕ ′′ ϕ + 16(5x − 2e 2 )ϕ+ +8(2x 2 − e 2 x − e 2 2)ϕ ′ + 64(2x 2 − e 2 x − e 2 2) 2 − 8[n(n +1)x − λ]ϕ}Z ′ 5+ + {24ϕ − 32[n(n +1)− λ](2x 2 − e 2 x − e 2 2)+ + 32(6x +3e 2 )(2x 2 − e 2 x − e 2 2) − 4n(n +1)ϕ}Z 5 =0, (V 1 ) 2ϕ 2 Z ′′′ 6 +3ϕ[ϕ′ +8(2x 2 − e 1 x − e 2 1 )]Z′′ 6 + {ϕ′′ ϕ + 16(5x − 2e 1 )ϕ+ +8(2x 2 − e 1 x − e 2 1)ϕ ′ + 64(2x 2 − e 1 x − e 2 1) 2 − 8[n(n +1)x − λ]ϕ}Z ′ 6+ + {24ϕ − 32[n(n +1)− λ](2x 2 − e 1 x − e 2 1 )+ + 32(6x +3e 1 )(2x 2 − e 1 x − e 2 1) − 4n(n +1)ϕ}Z 6 =0, (VI 1 )
ZA DIFERENCIJALNATA RAVENKA NA LAME 69 2ϕ 2 Z 7 ′′′ +9ϕϕ ′ Z 7 ′′ + {7ϕ ′′ ϕ +6ϕ ′2 − 8[n(n +1)x − λ]ϕ}Z 7+ ′ + {2ϕ ′′′ ϕ +4ϕ ′′ ϕ ′ − 8[n(n +1)− λ]ϕ ′ − 4n(n +1)ϕ}Z 7 =0, (VII 1 ) So koristeǌe na vrskite (4 ′ ) se dobivaat partikularni rexenija na diferencijalni ravenki od tret red od vid (I 1 ), (II 1 ), (III 1 ), (IV 1 ), (V 1 ), (VI 1 ), (VII 1 ), izrazeni so polinomnoto rexenie F na diferencijalnata ravenka (∗). Teorema: Diferencijalnite ravenki od tret red od vid (I 1 ), (II 1 ), (III 1 ), (IV 1 ), (V 1 ), (VI 1 ), (VII 1 ) imaat edno rexenie izrazeno so polinomot na Hermit F kako polinomno rexenie na diferencijalnata ravenka od tret red (∗). Pri toa formulata na soodvetnoto rexenie e dadeno so: Z 1 =(x − e 1 ) −1 F, Z 2 =(x − e 2 ) −1 F, yZ 3 =(x − e 3 ) −1 , Z 4 =(x − e 1 ) −1 (x − e 2 ) −1 F, Z 5 =(x − e 1 ) −1 (x − e 3 ) −1 F, (5) Z 6 =(x − e 2 ) −1 (x − e 3 ) −1 F, Z 7 =(x − e 1 ) −1 (x − e 2 ) −1 (x − e 3 ) −1 F. Literatura [1]E.Heine: Uber lineare Differentialgleichungen zweiter Ordnung, so wie uber die Existenz und Anzahl der Lame’shen Funktionen erster Art Monatsberichte der Kon Preuss. Akademie Wissenschaften, Berlin 1864 (1865), 13-22. [2] Boro Piperevski: One transformation of a class of linear differential equations of the second order, Proceedings, Department of Electrical Engineering, tome 6-7, (27-34), 1990, Skopje. [3] Sheffer, I.M.: On the properties of polynomials satisfying a linear differential equations: Part I; Transactions of the American Mathematical Society, <strong>35</strong>(1933), 184-214, Menasha – New York.
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05:35, 1 ÑÐ°Ð½ÑÐ°ÑÐ¸ 1970

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05:35, 1 ÑÐ°Ð½ÑÐ°ÑÐ¸ 1970