Question

please help in any of these questions in DE

Homework Problems for Handout Sheet 28 In Problems 1 to 3, determine the eigenvalues and the corresponding solutions for the given Sturm-Liouville BVP. 1. y" + 2y = 0, y(0)=0, y' (t)=0. 2. y" + 2y=0, y'(0)=0, y' (1)=0. 3. y" + 2y = 0, y'(0)=0, y(L)=0.

Answer 1

Date: 1 1 Page No. To y tiy=0 auxillary 910150 equation )=0 y' is caseo if i=0 > m=0.0 general solution is ya = AxtB. . y' 121= A using initial conditions y1o) = 0= A.O+ B. - B=0 we 3 get trivial solution Neo is not an eigen value (220) case . if yo. take de 22 mata2 = 0 - m?= -&? - ms Idi general solution is yol. A cosax +Bsinaa glW15 – Assinar + Bacosta using initial conditions.

UPH y10)= 0= Date: 1 All 1 Page No. GILAS S0 = B.O - A=0 Ba cosait B Cosat = 0 if B=0 then we get trivial solution cosato o - 7 = 1202 Aat2 - - Zo eigenfunction an eigenvalue is is f corresponding yW1= (constant) * sin (2ng)k, n=12 I case By x=-22 1970) if xco m2-420 amza 22 a masa genercel solution is y012 Apart Beta yleen= Age<*_ Ba é a using initiw conditions =) BE-A o AB yo - ६ ६२२ = 24 ९ ५' () = A ETT TAGET=0 a Aceart team) 20 =0 aaa coshat but coshan 70 solution an eigen value - trivial is not we get xco

RUPA Date: 1 Page No. e Auxiliary y101=0, y1=0 equation m2 + i=0 case o if d=0 - general solution is y01: ARAB ( from above examme) using given conditions 9101=0 = A.Q+B a B=0 y'(1 =0 A 3 A=0 we get trivial solution - x=0 is not an eigenvalue case ☺ if 180 X=22 1470) general solution is.. y1812 A cosax + Bsinda Yl60= -Aasinda + B2 cosas using given conditions. ( from Exo). 320120= AntB.0 = A=0 yli 20 = Ba cosa B cosa = 0 for non-trivia solution ke ( 24) cosa= 0 solution weget non trivial (224) n2 is an - TE eigenvalue eigen function.is. 3W constant) x sin (zmy na 121122

а Date: «5 18 тоо . - - - еекс 50\uКоо Кs (40) yWE AREX ylos - Beda Arete – Baeza (from Ex. 0 using given conditions. yol = 0 - 0 I'll ATB = 0 - B=-* O Age-BGER - - Aesta ET=0 - 2A Cosh = 0 but coshato we get xco 2 trivial solution is not an eigenvalue. © Y"+ xy = 0 , y10120, V'/L=0 Auxiliary equation is m + X=0 carse o if x=0 general solution is (from Ex. O) yGIE AXTB y = A using given conditions. - 1101=0 A:0+B=0 s B=0 yuko A =0 a 1=0 is eigenvalue not an lazo). =&2 case . if Azo d general solution is (from E.0) y01: A cosaxt Bsinza Aasinda + Ba costa

Date: H Page: esing given conditions: 01-0 =) A y'(L) to 8.0 -0 a med bacorab=0 a B cosal 20 Hor non-trivial solution = A= ( A=0) (250) a 22 = 122- 11 222 (224) 22 eigenvalue Mal2.- eigentünchen, is x= [ can no g is y01 = constants x sin lamene) n=1,2, cus ® е хъ general solution y = A esa g'ana Adera using given - - - (50) is Beta Baeza conditions y101=0 - A+B=0 =) BELA y'ch)=0 a Agent AREER 20 a 2A coshaa 50 but cosh Lx 70 A= 0 CB-A) = B= 0 Aco is not an eigenvalue