Download AKTU B-Tech 2nd Sem 2016-17 EAS203 Engineering Mathematics II Question Paper

Download AKTU (Dr. A.P.J. Abdul Kalam Technical University (AKTU), formerly Uttar Pradesh Technical University (UPTU)) B-Tech 2nd Semester (Second Semester) 2016-17 EAS203 Engineering Mathematics II Question Paper

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B.TECH.
THEORY EXAMINATION (SEM?II) 2016-17
ENGINEERING MATHEMATICS - II
T ime : 3 Hours Max. Marks : 100
Note .' Be precise in your answer. In case ofnumericalproblem assume data wherever not provided.
SECTION ? A
1. Explain the following: 10 x 2 = 20
(a) Show that the differential equation y dx ? 2X dy = 0 represents a family of parabolas.
(b) Classify the partial differential equation
(1 2 622 2 022 2 022 _
?x )W? xym+ (1?y )a?yz? ZZ
(c) Find the particular integral of (D ? 6023/ = e? f "(36).
(d) Write the Dirichlet?s conditions for Fourier series.
(e) Prove that 1'0(x) = ?]1(x).
(f) Prove that L [e??f(t)] = F (s ? a)
(g) Find the Laplace transform of f (t) = SimTat
(h) Write one and two dimensional wave equations.
(i) Find the constant term when f (x) = le is expanded in Fourier series in the interval (?
2, 2).
(j) Write the generating function for Legendre polynomial Pn (x).
SECTION ? B
2. Attempt any ?ve of the following questions: 5 x 10 = 50
(a) Solve the differential equation
(D2 + 2D + 2)}! = e?xsec3x, where D = %.
(b) Prove that (n + 1)Pn+1(x) = (2n + 1)xPn (x) ? nPn_1(x),
where Pn (x) is the Legendre?s function.
(c) Find the series solution of the differential equation
2 ? d_y _ _
2x dxz +xdx (x+1)y?0.
(d) Using Laplace transform, solve the differential equation
dzy _ . _ E _ _
?+9y ? c052t , y(O) ? 1, y(z) ? 1.
(e) Obtain the Fourier series of the function,
f(t)= t , ?TE< t<0
= ?t , 0 < t < 1r.
Hence, deduce that 112+ 312 + 5:2 + .. = ?
(f) Solve 62_u + azu =
6x2 W 0 under the conditions u(O, y) = 0,
nnx
u(l,y) = 0,u(x, 0) = 0 and u(x,a) = sinT.
(g) Solve the partial differential equation:
(D3 ? 4D2D?+ 5D D?2 ? 20?3)z = ey+2x + /?y + x
(h) Using convolution theorem find L ?1 [m]

SECTION ? C
Attempt any two of the following questions: 2 x 15 = 30
3.
(a)
Solve the differential equation (D2-2D+1)y = ex sin x
(b) Solve the equation by Laplace transform method:
:?:+2y + fotydt = sint, y(0) = 1.
(c) Solve the partial differential equation
(3/2 +zZ)p?xyq+zx = 0,wherep =:?: &q =:?:
(a) Find the Laplace transform of w
(b) Express f (x) = 4x3 ? 2x2 ? 3x + 8 in terms of Legendre?s polynomial.
(c) Expand f(x) = 2x ? 1 as a cosine series in 0 < x < 2.
(a) Show that 13 (x) = (9% ? 1) 11 (x) ? $1000.
(b) Solve the 2 3?: + 33?: + 52 = 0; z(0,y) = 2e?y by the method of separation of
variables.
(c) A tightly stretched string with fixed end x = 0 and x = l is initially in a position
given by y = using . If it is released from rest from this position, find the
displacement y(x, t).