Download Mathematical Equations and Formulas and more Exercises Mathematics in PDF only on Docsity! Dale Tulodal2 2s lotb Unit-T Apcorlons of pte fovmulas CDT-4 One-Dirmensibra Wave E?uotion (vilbrallns ofa stolched dorg) ue equation fos 04 il o Boundary Conditors yCO)=0 y (lD-0 Infla Cordtlins f y(z0)=0 a/0-f()-) ot Genesol Soluifo crit 4bo Col yCxyt)= Stn- a,h + boGat Sufloble Solutfon yCx= (apa +C2shpa) (Gcacpl c4sincpt CDT-5 One-Dinmendona teat -fto0 uOale e9 uatfon Ou-dou ot Baurdoy Candionss u Cot)=o u (it)=O u (z/0) =f) Suttable Solutfoni u Cxt)=Ca CosprC sinp) PE c DT-6 Hect flou though a ay of firte length ron hoogeneaus boundary Condittons. Ou u Suitable seutfon ulat)= Ca Coap +stopa) P y(zt)= sin Cos nTE step Substftuting 6) n ), toe qal y(z0)-[ (z,+-0 Jt-O COAt s Cas C put n=1 ). 3 sin sofi))_nb Sin 6, Sin 2 b 3in Saa. - Cowpau tike terins b39 by 0, b3- b4-br o b Sin cosht+b, Sin 3ha tos C3t 31 Cos 37E -yo sto 3Ho gfoTx_Cos 4 Ofunlt 3) find he defle elfoan of a vattrg Song unl length haing fred eds with bflta Velocty am and totttal defle ctton (a)- a (x- *) () ou Ox y(o.o)-0- y (1,0) 0 ou ot y(xo)= a (x- 2) 6) y pinCple of Qperpoc tfor0, The gncIal soluktbn yxt)- Sin x an Cos(Mct+br, Sih/Mct OMct bo Cos cE ot OAC CnT Du stin an o-bov ot cO b put bnrO fo e2n /DMct yCt) gC%o) n sin/M.)-a (z- ) (x)ado(nM) x Cas( nn) M (on) a sto(on)d 9a-). -casl nnx) Sh -n O ao-1 Cos(nx foDn Con O (om L a -a- 40(1-(-1)") ® put e2n h y(xA)= aa(1-(-)) sto (Om) Cos (nact (nt y(x )= Substftute u (10,0)= 100 100 a (10)4 5010 a:5 ulz,0)- 5x+50 6 Now ot u (ot)= 90-(6) u C10,t)= 60>G) The steody Stocte tempexatuse distibutibn UsC)= uCot) + | u(107t)-u[ot) 90+ /Go-90 10 0+ )x Us (= 90-3x (8) The tanslerst tempeyathuve dist olbttfoy -ept u(xA)- on nCospz + bnsfo Px) e -9 he teropeahee u Cart) fro -the mteamecliate pe sbo & u(xt) Us ()+ ut (z,-4) -p u Cxt) - 90- 3x 4E( Cotpx +bnshPX)e (10) sep Subst tute (6) to ao) u C 90 0-82 +Sanospx + bo sinpx) eP 90 an+ bn(O)=0 on- O L0e get put an = O h 10 u CxA)= 90-3x o e SioPx step-2 Subsitute C)o (U) =60 CuCxt) t sopx 6o 90-3x + =0 10 en +0 Sinp(10)= 0 = O 10p-n 10 Rt p D u, we get 10 sinp e u Cxrt= 0-31 +S b cot bo u CxA) o-3x4 n 10 O=1 step-3 Substthte 6) 12 uCzA-5x4 50 COA oS Stn10 e Sx+50 = 0-3x + tO 5X+50 90-3x + b sin O 10 8-40- so -fa) fouste Seste S: b2-Pcx) Sn e
