Download mce 101. analytical methods in engineering

M.TECH. DEGREE EXAMINATION
Branch : Civil Engineering
Specialization – Computer Aided Structural Engineering
Model Question Paper - I
First Semester
MCE 101. ANALYTICAL METHODS IN ENGINEERING
(Regular – 2011Admissions)
All questions carry equal marks
Time: 3 Hours
Total: 100 marks
1. Solve the following differential equations.
(a) ( D2 – 3D + 2 ) y = x e 3x + Sin 2x
(b) ( D2 + 2D + 1) y = 2x + x2
(c) x2y ″ + xy′ + y = log x. Sin ( log x)
OR
2. Solve using variation of parameters.
(a) ( D2 + 4 ) y = 4Sec22x
(b) ( D2 + n2 ) y = Sec n x
(c) x2y″ + 2 x y – 20 y
= x4
3. Solve the following equations.
(a)
dy
dx
dz


2
2
2
2
x( y  z ) y ( z  x ) z ( x  y 2 )
2
(b) ( D2 – 2 DD′ + D′ 2 ) z = e x + 2y
(c) x p + y q = z
OR
4. Solve using Charpits method.
(a) z2 = pqxy
(b) xyp +yq + pq = yz
[ P.T.O]
2

2  
a
5. (a) Verify that φ = e Cos x is a solution of the heat equation.
t 2
x 2
2t
for suitable ‘a’
(b) Find the integral surface of x2p + y2q + z2 = 0 ; which pass through the hyperbola
xy = x+y, z=1
OR
6. A plate occupies the semi infinite strip 0 ≤ x ≤ 1, 0 ≤ y ≤ ∞ ; The edges x = 0, x = 1 are
insulated. The edge y = 0 is kept at temperature f (x) = x (1-x). Find the temperature
distribution in the plate.
7. Solve the Laplace equation for the region bounded by the square 0 ≤ x ≤ 4, 0 ≤ y ≤ 4; the
boundary conditions being u=0 at x=0 and u = 8 + 2y at x = 4. u = x2 when y = 4 .
Take h = k = 1.
OR
8. Evaluate the pivotal values of the equation utt = 16 uxx, taking Δx = 1, up to t = 1.25. The
boundary conditions are (1) u (0,t) = u ( 5, t) = 0
(2) ut (x,0) = 0
(3) u (x, 0) = x2 (5-x) , 0 ≤ x ≤ 5
(4 x 25 = 100 marks)