Sun jeic Diaforikèc Exis¸seic II

24 IounÐou 2010

1. (3m.) DÐnetai h sun jhc diaforik  exÐswsh (1− x2)y′′(x)− 6xy′(x)− 4y(x) = 0

(aþ) Na brejoÔn ìla ta an¸mala shmeÐa kai na qarakthristoÔn. Aitiolog ste

(bþ) Na brejeÐ h genik  lÔsh me th mèjodo twn seir¸n se perioq  tou apeÐrou

(gþ) Na diereunhjeÐ h sÔgklish twn seir¸n-lÔsewn

2. (1.5m) DÐnetai to sÔsthma Sun jwn Diaforik¸n Exis¸sewn:

(D − 1)x(t) + (D + 1)y(t) = t2 , D2x(t) +Dy(t) = t , ìpou D = d/dt

Na lujeÐ me th mèjodo thc apaloif c

3. (2m.) JewreÐste thn exÐswsh tou mh grammikoÔ talantwt :

x′′(t) + βx′(t) + x(t)− [x(t)]2 = 0 ,ìpou β ≥ 0 suntelest c trib¸n.

(aþ) Metatrèyte thn parap�nw exÐswsh se sÔsthma sun jwn diaforik¸n exis¸sewn pr¸th-c t�xhc kai breÐte tic sunj kec isorropÐac autoÔ

(bþ) Melet ste thn eust�jeia aut¸n kaj¸c h par�metroc β metab�lletai. JewreÐstearqik� β = 0, met� 0 < β < 2 kai tèloc 2 < β

4. (2m.) DÐnetai to prìblhma sunoriak¸n tim¸n

y′′(x) + (1− λ)y(x) = 0, y(π) = 0, y′(0) = 0, 0 ≤ x ≤ π

(aþ) Na brejoÔn oi idiotimèc kai oi antÐstoiqec idiosunart seic tou parap�nw P.S.T.

(bþ) Na exetasteÐ an eÐnai P.S.T. tÔpou Sturm-Liouville. EÐnai oi idiosunart seic touorjog¸niec sto di�sthma [0, p] wc proc th sun�rthsh b�rouc ρ(x) pou up�rqei sthnexÐswsh;

5. (2m.)

(aþ) Na upologisteÐ o metasqhmatismìc Laplace thc sun�rthshc f(x) = 1 kai na dojeÐto di�sthma Ôparxhc autoÔ. Me autì wc dedomèno, na dojeÐ o metasqhmatismìcLaplace kai to antÐstoiqo di�sthma Ôparxhc gia th sun�rthsh g(x) = eax

(bþ) Na lujeÐ me qr sh metasqhmatismoÔ Laplace to Prìblhma Arqik¸n Tim¸n:

y′′(x)− 3y′(x) + 2y(x) = e3x, y(0) = 1, y′(0) = 0