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J L278B.E./8.Tech.DEGREE EXAMINATION, MAY/JUNE 2006.

Third SemesterMA 231 - MATHEMATICS - III

(Common o all branchesexceptBiomedicalEngineering, Civil Engineering andComputer BasedConstructions,FashionTechnology, ndustrial Bio-Technology,Textile Chemistrv)fime : Three hours Maximum: 100marks

Answer ALL questions.PARTA- ( fO x2=2}marks )

Form partial differential equation by eliminating thes = f(xy). arbitrary function from

2. Write dorvn he completesolution of z = px,+ qy + c

Find o" in expanding e-" as Fourier series in (-n, tt ) .

State Parseval's Identity of Fourier series.5. A tightly stretched string of length 2 L is fastenedat both ends. The mid pointof the string is displaced to a distance '6' and released from rest in thisposition. Write the Initial Conditions.

In onedimensional heat equation ut =a2 ao . What does a2 stands for?State initial and frnal value theorems.Define convolution and convolution heorem of Laplace transforms.rf F lfbc)j = l(s) then give the value of F {f@x)\ .

3.4.

6.7 .8.9.

10. Find Fourier transform of f(x)- 1 l" | = t-0 l r l r t .

L+p '

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PARTB- (5x16=80marks )11. (i) Solve x -22)p+(22 - y)q = y -:(

( ^ _ . " 1(ii) Solve \D' * 4DD' - 5D'' I z = sin(2x + 3y) '

L2. (a ) (i) Expand f(x) = x' - x as Fourier series n (-t,tr) '(ii) Find Half Rangecosineseriesgiven

f ( x )=x 0 ( r

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(br (i) Using convolution heorem ,t {---:-}l (s" + a" )"( i i . r So i ve * *=cos /dt

dx_ . r y =s in / r (0 )=2 y (0 )=0 .d.tt215. (a) (i) Find Fourier transform of e-o"' , Hence prove e- 2 isself reciprocal.

(ii) Find Fourier Sine and Cosine ransform of x"-r.Or

(b) (i) Using Parseval's Identity for Fourier cosine transform of e-o'evaluate' l , uo* u, , .d \ a ' + x ' f

(ii) Find Fourier Sine transform of e-"* (o > 0). Hence ind F" kr-".|.

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