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Mathematical Equations and Formulas, Exercises of Mathematics

A series of mathematical equations and formulas related to one-dimensional wave equations, boundary conditions, and temperature distribution. It also includes a problem-solving example related to the deflection of a vibrating string. suitable for students studying advanced mathematics and physics.

Typology: Exercises

2022/2023

Available from 03/08/2023

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venky-s-1 🇮🇳

5 documents

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bg1
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
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe

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Dale Tulodal

2slotb

Unit-T Apcorlons^

of pte fovmulas CDT- One-Dirmensibra Wave^ E?uotion (vilbrallns ofa^ stolched (^) dorg) ue equation fos 04 il o Boundary Conditors yCO)= y (lD- Infla Cordtlins f y(z0)= a/0-f()-) ot Genesol Soluifo crit 4bo Col yCxyt)= Stn-

a,h

boGat

Sufloble Solutfon yCx= (apa +C2shpa) (Gcacpl c4sincpt

CDT-

One-Dinmendona teat -fto uOale e9 uatfon

Ou-dou

ot

Baurdoy Candionss

u Cot)=o u (it)=O u (z/0) =f)

Suttable Solutfoni

u Cxt)=CaCosprC sinp) PE

c DT-

Hect flou^ though a^ ay^ of^ firte^ length^

ron hoogeneaus boundary Condittons.

Ou u

Suitable seutfon

ulat)= CaCoap +stopa)^

P

he bournday Cedtlfons^

ae

y

(o,t)-o-> ond yClt)-0- -O ot-

Also the tottial veloctty

and (^) yo)- Ye^ 5to )^ -> y

the pincide of^ supexpodtton,^

the general

Sdutfo of (1) ts Cont cnMtbo Cas yat)= E^ Sto-X.(an Sto^ bo Cas^ ONE step Substthctfog (4)fnG) , get

Oy

ot

CrTt contan Cos 0 sthZan sin O- Sto AnCos COE-bnsonE/t t- O-0 (^0) stn (o|.o n

pat

an-o fo^ 6),^

uwe pt

y(zt)=

sin Cos^

nTE

step

Substftuting6) n ), toe qal y(z0)-[ (z,+-0Jt-O COAt

s Cas

C put n=

sin sofi))_nb (^) Sin (^) 6, (^) Sin 2 b (^) 3in Saa. (^) - Cowpau

tike terins

b by (^) 0, b3- b4-^ br^ o

b Sin cosht+b, Sin3ha tos C3t

31 Cos 37E -yo sto 3Ho gfoTx_Cos

(x)ado(nM)

x Cas(nn)^ M

(on)

a (^) sto(on)d

9a-). -casl nnx) Sh -n

O

ao-1 Cos(nx^ foDn O (^) Con (om L a -a- 40(1-(-1)") ® put e2n h y(xA)= aa(1-(-))sto^ (Om)^

Cos (nact

(nt

y(xA)=

  1. 1D heat flao epuatbn wth al Define Condrtfons. Also explain^ Hs^ Sdutfons, pocsfble (^) Sdudions CuaNe (^) eguation fox AD he^

at Ploo

du a Du ot the (^) Boudary candilbos^ ale u (o,t)=o > u (t)^ =0^ > ho (^) fnitio tempexctuse distibcfon h^ the^ kar Tho s ulzo)= P(X)>O The Vasious^ pocsble^ soludtfons (^) oF the^ heat Plouo (^) euaton by^ vahble Sepexable methomethao Sepesable cpt uCzt)= (1eaG a)ePt

uCxt) =^

(c4Cespz +^ C5^ npz)^ ee

(Ety uC7/t)=^

(caX +Cg)Cq

The (^) only suttablo^ Solutfon s pt

u CxA)=^ Ca^

Catpx +Cs^ dopx) g^ e

Substftute u^

100

a (10)4 50100100

a:

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- 10 0+ )x Us(=^ 90-3x^ (8) The tanslerst^ tempeyathuve^ dist^ olbttfoy

-ept

u(xA)- onnCospz + bnsfo Px) e (^) - 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

Substttute^ (6)^

to ao) u C 90

+Sanospx +^ bo sinpx) eP^ 90 an+ bn(O)=

on- O

L0e get put an^ = (^) O h^10

u CxA)=^

90-3x

o e^

SioPx step- Subsitute C)o (U) = CuCxt)

t sopx 6o

90-3x +

10

onX, dx

Px) sn 10 (8x-46) sfo ol 40 U 8-40 (^) V sP^ NA

u-

V= 10 Cos 10 Va 10 sf n2 10 10 2 (sx-46)^ (0. cos (^) +s s -(4) 10

)P

  • ( 4o 40 10 ft-a(10 41 10 bn SO -+ Substftcte b^ 12)^

e g t

u Cxt) = 9o-3 +

  • (1-+-1° )tsi 0 e

uCxr- (^) q0-3x - So(14G1)TO

sfnnT

10

6) Find^ the^ tempercdure destibution^

fo tho bar act lime t fr the poblem ()