Question

Question 3: BVP with periodic boundary conditions. Part I: Solve the following boundary value problem (BVP) where y(x,t) is defined for 0<x<. You must show all of your work (be sure to explore all possible eigenvalues). агу д?у 4 axat2 Subject to conditions: = y(x,0) = 4 sin 6x ayi at = 0 y(0) = 0 y(T) = 0 Solution: y(x, t) = Do your work on the next page. Part II: Follow up questions. You may answer these questions with a simple equation or sentence. Do not do any extensive solving here: a). What mathematical technique or tool would you use if the initial position was y(x,0) = 4 cos 6x? b). How would you solve the problem if both position and velocity initial conditions were present? y(x,0) = 4 sin 6x ду at y(0) = 0 y) = 0 = G(x) t=0 c). What spatial solutions would be solve the system if the boundary condition was changed to ду| = 0 & = 0 at at ayl X=0 X

Answer 1

Angwer at2 4824824 282 subject to conditions: y (20) = 4 sin 68 1100 146)-0, 4(1)-0 Take finite fourier sine transffim of 0 weget. 1984 cpnsrede = sinsr dr. at2 a82 d² T dt [88054 ay an J'y or ords=14 T-05-06 Sedy de di 2 y = 0-48 = 0-43 / cos se ay de = -45 {(cosse.yt to jsinsa.yde. = 0-482 Чо {:4(0)-0,4)-04 d²y + 4 g2 yg=0 dt2 Its A.Eis De + 4 g2=0 D-APRS . Its solution is ý (sit) = acos 2st + Cesin ast Now, put t=0, weget gg (510) a fg{y (810)} =ci

Sty (810) sin sse de-a i 48iner sin grede=cn 2&siner einse dreza C.RSInASiDB = COS (A-B)-COS (A+B)] 2/ (cos (6-8) * 3) X-COS (6+9) 2]d8-C sincase stot 49) SROCE_9)T_$oCG+9)]- 6-8 64S =G 648 singinosis-cosensions sing costist cost slots 613 = 2 To - gints. I 6-8 841. gints 649 F2 [-sions Sints 6+3 6-8 -P (SAS) [6 6+5 - 2 (Sints) / 6+6+68 36-82 -bolsin (13) 12 36-g2

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