Question

Solve the following wave partial differential equation of the vibration of string for ?(? ,?).

y_xx=16y_tt

y(0,t)=y(1,t)=0

y(x,0)=2sin($\pi$x)+5sin(3$\pi$x)

y_t(x,0)=6$\pi$sin(4$\pi$x)+10$\pi$sin(8$\pi$x)

Answer 1

y (0-4) = y(4.4) Boundary condition Alo antial condition 4(a.o) = avintat 5 sin Bra) Givan that, You = 16 Yff su acău 50 (L1) ac an ,: č=1 Yq (no) = 6tt sin(4TTU 1+ loftsin (8ra) » / 2 = 1 4 (6 We know that Ware equation for defefion y(art) = Ê (Bn comht + Bo* sin apt] sin qua where an = nitc OT] 4x1 - MT 4 r! y Curt) - [Bn conne + bot sin ng tsin min By Applying initial conditionsielt=oj yenro) = (En co, ng@+ cong @1 zi Jippalen ? sin noin Yaro) E Bn sinn istituting y(«.0) from given Quinita + 5 sin 3TTA = B, Sin Tu + Bu sin2tra +33 sin 374 n1 NET "Bu sin unut Butsinnta By comparing By=2; B2=0 i B3=; Bu zBg - Poem - 3Bn0

SO By chifferentiating 46.45 w.vit tinue to givevelocistas y (out) = f (3. –sin Ant (an) + Bo costant.ro] sin mua ft intial velocity 94 610) Ye Chro) = ( On-sien Gablow to Buy cos Koco). 6m) xin nga 6 n=1 YqCno) SEO (Bran] sin ma nal Anant 4 i 67T sin(40W) + ron singt i = (B. I sinton + B ) sin ain Bi no sin nin + By comparing on both sides n=1 B =0 ; B4 = 6 i Bg5; and allother valves = 0 de B₂ B3 = By Ri=0 since y lurt) ★ (Bn los ant to sinant] sin nan given ..tette fecom B23001 sat fesinfitussinmt y(t) = 2003 TTt sin fin + 300 30 stytsin tu + asintesinoita + 5 sinantsinsta प Thank You ☺
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