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Quiz3_Spring_2010_Solutions

# Quiz3_Spring_2010_Solutions - constant at 0oC Find the...

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rL I J_L r.L t J-l ../ 869M. ----_-- Zo 9"'* MAE 105: Introduction to Mathematical Physics QUIZ 3: April30,2010 I Useful formulae: i "or(^*)sin(nd)dx = 0 J, L' L' L Show, tttut i sin(!Ey* = I t^ (.}.) dnt- = as (a3r;1.* : /-r^ A!9 2. When a function, f(x) is expressed as a Fourier series over an interval (-L, L), as : ao +f a n cosl!91 * t4 sn(ry) , show that b, = ! \ I I ft,t"i*!g uo -f;oun fvus(-/. *r"" sur(-,| , ouvyv .ttsL un: i ),f (x)sin(-:::px tJtr. s,&" '6 1l+) aA {p* uri^"ot: "{q)& : "" J_i i*(u}*-),1" "i o"l',;,("ry) s^(q)dd r-L L - --: = O j.'*^ .f.'#l 3. Consider heat conduction in a one-dimensional rod of lensth 7T caused bv a heat source 1 Q @, t) : T osinr, where T ois aconstant. Both ends of the rod are submerged into cooling baths, which maintain their temperature
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Unformatted text preview: constant at 0oC. Find the steady state temperature profile inside the rod, I(x) usei ary;il=oz o2\:,t) [email protected],t) 4 4L A"T(',t) 1 6-h,t) = OxL fi b,Ir."rJo^A a"xJshu5 T (t= o)= o U ftt- nt"^\$ s'r?k E-I!'') : a at Name: SID: lsle.: c.s(_O).(-s c r- r .-1- (,os(-rnTi)l=n J 7 boL tf r.Js; (y)*(N** Ali -L 't A"r("1 + &h) L^('n! =o _lr*t ry* : 0' if m is an integer'-I [",(^q")l^ = -j fagn) LTl-L ' I Jl-t- 7nTr, L f *^1^ru17' : -L" f.,.(^rr)-*rrL '--Lt-, ;TIL @rb v*v f o"l n--, q f(x) t4trLt\ fbT + J'+r*l-L w )i A= dtt" * rh=fr/r) f l6 7 Suix < E-d.-4 V"It*alt * t''st-"--W fb) = !;S'""t *&'xt aY L 1 \$r(*) * = --G 2 o<z 7T d1 T (x) = ft.--Sw,x - 7; Z. x b<- x^ lT I T[x---o) - o -) f(x=fr7r)=o 9 s^(n6)"r o, F/r)=o...
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