Download AKTU B-Tech 4th Sem 2016-17 RAS203 Mathematics Ii Question Paper

Download AKTU (Dr. A.P.J. Abdul Kalam Technical University (AKTU), formerly Uttar Pradesh Technical University (UPTU) B-Tech 4th Semester (Fourth Semester) 2016-17 RAS203 Mathematics Ii Question Paper

Printed Pages:2 Roll No.| l l l l I l l l l l RAS203
B.TECH.
THEORY EXAMINATION (SEM?IV) 2016-17
MATHEMATICS-II
Time : 3 Hours Max. Marks : 70
Note : Be precise in your answer. In case Ofnumerical problem assume data wherever not provided.
SECTION ? A
1. Attempt any seven parts for the following: 7 X 2 = 14
2
(a) Solve the differential equation % = ?12x2 + 24x ? 20 with the condition X = 0, y =
5 and X = 0, y = 21 ad hence find the value of y at X = 1.
(b) For a differential equation % + 2053?: + y = 0, find the value of 0c for which the
differential equation characteristic equation has equal number.
(c) For a Legend polynomial prove that PH (1) = 1 and PH (-1) = (?1)n
(d) For the Bessel?s function Jn(X) prove the following identities:
J_n(X) = (-1)"]n(x) and I?n(-X) = {-1)"]n(x)
(e) Evaluate the Laplace transform of Integral Of a function L { I; f (t/ dt)}.
(f) Evaluate the value of integral L? t. e'Ztcost dt.
(g) Find the Fourier coefficient for the function f (x) = x2 0 < X < 2n
(h) Find the partial differential equation of all sphere whose centre lie on Z-aXis.
(i) Formulate the PDE by eliminating the arbitrary function from ?(x2 + 312.372 + Z 2) = 0
(j) Specify with suitable example the clarification Partial Differential Equation (PDE) for
elliptic, parabolic and hyperbolic differential equation.
SECTION ? B
2. Attempt any three parts of the following questions: 3 X 7 = 21
2
(a) A function n(X) satisfies the differential equation (1%?) ? g = 0, where L is a
constant. The boundary conditions are n(0) = X and n(00) = 0. Find the solution to this
equation.
(b) Find the series solution by Forbenias method for the differential equation
(1 ? x2)y" ? 2xy? + 20y = 0
(c) Determine the response of damped mass ? spring system under a square wave given by
the differential equation
y" + 3y?+ 2y = u(t ? 1) ? u(t ? 2), y(0) = O, y?(0) = 0
Using the Laplace transform.
(d) Obtain the Fourier expansion of f (x) = x Sin x as cosine series in (0, 7t) and hence
show that
1 1 + 1 _ (7r ? 2)
1 X 3 3 X 5 5 X 7 ? In. In. In. In. I ? 4
(e) Solve by method of separation of variable for PDE
a_u 2_ _ ?x
xax+Zay?0, u(x,0)?4e

SECTION ? C
Attempt all parts of the following questions: 7 x 5 = 35
3. Attempt any two parts of the following:
(a) Find the particular solution of the differential equation
dz_y 1 _
M + a ? sec ax
2
(b) If y = 31106) and y = y2 (x) are two solutions of the equation % + P(x) 2?: +
Q(x)y = 0, then show that yl (%) ? yz (%) = ce?f de, where c is constant.
(c) Solve by method of variation of Parameter for the differential equation :
d2 y
@?6:?:+ay=(:::)
4. Attempt any two parts of the following:
(a)
(b)
(C)
Prove that E.]3/2 (x) = (i sin x ? cos x)
Show that Legendre polynomials are orthogonal on the interval [-1, 1]
Prove that [:11 xPn (x) dx = ?
5. Attempt any two parts of the following:
(a)
(b)
(C)
Find the Laplace transform of SRw ? tooth wave function
F(t) = Kt in 0 < t < 1 with period 1
4
$2 + 25 + 5
Solve the simultaneous differential equation, using Laplace transformation ?
:_:I+2x=sin2t; :?:?2y=c052t, WhereX(0)=1,y(0)=0
Use Convolution theorem to find the inverse of function F (s) =
6. Attempt any two parts of the following:
(a)
(b)
(C)
(a)
(b)
n?x 2 11:2 00 1
Iff(x) = T]? 0 < x < 21: then show that f(x) = E- Fl?cosnx
Find the complete solution of PDE
(A2 + 7AD' + 12D?2) / 2 = sin hx, where symbols have their usual meaning.
Solve the PDE p + St] = 52 + tan(y ? 3x)
Attempt any one part of the following:
A square plate is bounded by lines X = 0, y = 0; X = 20, y = 20. Its faces are insulated.
The temperature along the upper horizontal edge is given by u (X, 20) = X (20 ? X) when
0 < x < 20 while the upper three edges are kept at 0?C. Find the steady state
temperature.
A bar of 10 cm long with insulated sides A and B are kept at 20?C and 40?C
respectively until steady state conditions prevail. The temperature at A is then suddenly
varies to 50?C and the same instant that at B bowered to 10?C. Find the subsequent
temperature at any point of the bar at any time.