MATHEMATICS-II

UTTARAKHAND TECHNICAL UNIVERSITY

B.TECH (FIRST YEAR)

2012

time:- 3 hr

Total Marks:100

Section A

Note:- Attempt all questions. All question carry equal marks.

Q1 Attempt any four parts of the following:-

5x4=20

1. Solve:
   [y²e^xy² + 4x³]dx +[2xye^xy²- 3y²]dy =0
2. Solve:(1+y²)dx =(tan^_1y –x)dy
3. Solve:
   (D²-2D+1)y = xsinx
4. Solve:
   
   dx/dt + 2x -3y = t, dy/dt -3x + 2y =e^2t
5. Solve:
   
   d²y/dx² + cotx dy/dx + 4y cosec²x =0
   by changing the independent variable.
6. Apply the method of variation of parameter to solve:
   d²y/dx² +n²₄ = secnx
   
   Section B
   

Q2:- Attempt any two parts of the following :

10x2=20

1. Find Laplace transform of
   (a)  te^at sin at
   (b) (cos at – cos bt)/t
2. Find
   (a) L^-1[log S+1/S-1]
   (b) L^-1 [1/S(s+a)³]
3. Using laplace transform solve the following equation:
   d²y/dt² + x = t cos2t
   given x(0) =x^’(0)=0

Section C

Q3:-Attempt any two parts of the following

10x2=20

 

 

(b)Test for convergence the series

    

     2.  Discuss the convergence of the series

 

     3. Find the geometric series.

Section D

Q4:- Answer any two parts of the following:

10x2=20

1. Find the fourier series expansion for f(x) ,if
   
     
2. f(x)= expand this as Fourier sine series.
3. Solve (D³-4D²D + 4DD^’2) = 6 sin(3x+2y) 

Section E

Q5:- Attempt any two parts of the following :

10x2=20

1. A tightly stretched string with fixed end points x=0 and x=1 is initially in a position given by y(x,0) =  y₀ sin(πx/l). it is released from rest from this position, find the displacement y at any distance x from one end at any time t.
2. A homogeneous rod of conducting material of length 'l' has its ends kept at zero temperature. The temperature at the center is T and falls uniformly to zero at the two ends. Find the temperature distribution.
3. Solve     given u(0,y) = 4e^-y – e^-5y, by the method of separation of variables.