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Posted Date: 22 Oct 2009 Posted By: Gourav Kumar Satpathy Member Level: Bronze
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2006 Biju Patnaik University of Technology B.Tech Computer Science & Engineering mathematics-3 Question paper
THIRD SEMESTER EXAMINATION 2006
MATHEMATICS-iii
Question 1
(a) When is the equation of the form
AUxx+2BUxy+CUyy=F(x, y, U, Ux, Uy) said to be parabolic, elliptic, or hyperbolic?
(b) Find the solution of the equation
XUxy+2yU=0
By the variable separation method?
(c) Show that the function f in U(x,t)=0.5[f(x+ct)+f(x-ct)] with boundary conditions U(0,t)=0, U(L,t)=0 for all t, is odd and is of period 2L.
(d) Show that the solution of the Laplace equation in spherical coordinates dType equation here.epending only on ‘r’ is given by U=c/r + K
(e) Find the fixed points of the function
F (z) = (z-1)/ (z+1)
(f) Find the cross ratio of the numbers 1,-1,I,-i
(g) Find all the values of (-1)1/4.
(h) Find the points at which the function
F (z) =cosecz
Fails to be conformal.
(i) Find the residue of the function
At its pole(s).
(j) Find the value of
Question 2
(a) Find the temperature u(x,t) in a laterally insulated copper bar 80cm long if the initial temperature is 100sin( ) 00 C
And the temperature ends are kept atO C copper C=1.158 cm/sec].How long will it take for the maximum temperature in the bar to drop to 500 C? (5)
(b) Transform the equation
Uxx -4Uxy +3Uyy=0
Into normal form. (4)
Question 3
(a) Find the steady state temperature U(x,y) in square copper plate with side a=30cm the faces being perfectly insulated; the upper side is kept at 40 c and the other sides are kept at 0c.(6)
(b) Find the deflection U(x,y,t) of the square membrane with a=b=1 and c=1 if the initial velocity is zero and the initial deflection is 0.1sin(3px)sin(4py).(4)
Question 4
(a) Find the potential in the interior of the sphere with R=1 assuming that there are no charges in the interior and the potential on the surface is f(?) =cos2?.(5)
(b) Solve the following differential equation using Laplace transforms Ux+2xUt =2x with boundary conditions u(x,0)=1=U(0,t).(5)
Question 5
(a) Given u(x,y)= (5)
(b) Find the linear fractional transformation which maps I,1,and –I to 0, -I and 8 respectively.(5)
Question 6
(a) Find the image of the unit circle around the origin under the mapping(4)
(b) Expand the function f(z)=
In a Taylor series around the origin. (2)
(c) Expand the function
In a Laurent series valid for z<|z|<3. (4)
Question 7
(a) Find the values of the following integrals:
(i)
(ii)
Where C= {z: |z|=2}.
(b) Find the poles and the residues at the poles of the function
Question 8
(a) Use the residue theorem to evaluate the following:
(i)
(ii) (5+5)
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