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A2801 Pages: 2

Module III
Obtain the Fourier series expansion of f(x) = x sinx in the interval (-7, 7). (6)
Find the half range sine series of f(x) = k in the interval (0, 7). (5)
OR
5 Fouri F _ (oxy, | (6)
Find the Fourier series of f(x) = (=) in the interval (0,272).
Find the half range sine series of f(x) = e* in (0,1). (5)
Module IV
992952 az __ ~) (6)
Solve a 4 ನಾ + 4 ‏روج‎ = 2 sin(3x + 2y)
Solve x(y? — 25)ற + y(z? — x?)q = 2062-35) (5)
OR
2 2 2
Form the PDE by eliminating a, 6, c from ಸ + ಶ್ಯ + = =1 (6)
Solve (x + y)zp + (x —y)zq =x? + y?. (5)
Module V

A tightly stretched violin string of length ‘a’ and fixed at both ends is plucked at (10)
its mid-point and assumes initially the shape of a triangle of height ‘h’. Find the
displacement u(x,t) at any distance ‘x’ and any time ‘t’ after the string is released
from rest.

OR

ந 2५ _ (२००४ (10)
Solve the PDE 55 = ० दद्ध

Boundary conditions are u(0,t) = u(l,t) = 0, ‏ع‎ 20
Initia] conditions are y(x,0) = asin (೫) and a =0 at t=0.

Module VI
A rod, 30 cm long has its ends A and B kept at 20°C and 80°C respectively, until (10)
the steady state conditions prevail. The temperature at each end is then suddenly
reduced to 0°C and kept so. Find the resulting temperature function u(x,t) taking
x=0 at A.

OR

A long iron rod with insulated lateral surface has its left end maintained at a (10)
temperature 000 and its right end at x=2, maintained at 100°C. Determine the

temperature as a function of ‘x’ and ‘t’ if the initial temperature is
56 0) = [10०*. 021
/ ~ (100 , 15252

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