FirstRanker.com
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER- ITI(OLD) EXAMINATION - SUMMER 2019
--- Content provided by FirstRanker.com ---
Subject Code: 130001 Date:30/05/2019
Subject Name: Mathematics-I11
Time: 02:30 PM TO 05:30 PM Total Marks: 70
Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
--- Content provided by FirstRanker.com ---
Q1 (a) Obtain series solution of dy/dx + y = 0. 07
(b) Attempt any two of the following. 07
- dy/dx = x(2logx+1)/siny+ycosy
- xdy-ydx=v(x2+y2)dx
- dy/dx + ycotx = cosx
- sin(y-xp)=p . Where, p = dy/dx.
--- Content provided by FirstRanker.com ---
Q.2 (a) Obtain the Frobenius series solution of 2x2y" +3xy' — (x2 + l)y =0. 07
(b) Attempt any two of the following. 07
--- Content provided by FirstRanker.com ---
- d4y/dx4 +13 d2y/dx2+36y=0
- (D2+3D+2)y=5.
- y" -2y' +y=cos2x
- Solve by Method of variation of parameters. y" +a2y = tanax.
OR
--- Content provided by FirstRanker.com ---
(b) Attempt any two of the following. 07
- x2 d2y/dx2 -x dy/dx -By=x2log x.
- Using method of undetermined multipliers solve y" +4y =8x2.
- Using method of undetermined multipliers solve y" —3y' +2y =ex.
- Prove that ?08 xn e-x dx = G(n+1)
--- Content provided by FirstRanker.com ---
Q.3 (a) Define Laplace Transformation of a function f (t) and using it obtain L(sin at) and L(tn). 07
(b) Attempt any two of the following. 07
- Find the Laplace transform of e-3t +sin 2t + 5cosh2t.
- Evaluate L(sin t sin 2t sin 3t).
- Evaluate L[e-t (cos4t +3sin 4t)].
- Evaluate L-1{ 1/(s-1)(s2+1) }
--- Content provided by FirstRanker.com ---
OR
Q4 (a) State Dirichlet’s conditions for Fourier series of a function f (x) and obtain Fourier series of a periodic function having fundamental period p. 07
(b) Attempt any two of the following. 07
--- Content provided by FirstRanker.com ---
- Evaluate L{ 1-cos2t/t }
- Evaluate L(t3e-3t)
- Evaluate L-1{ 1/s(s2+a2) }
- Using Laplace transform technique solve the following IVP. y" +4y=sin t, y(0)= 1, y'(0)= 0.
Q4 (a) Using convolution theorem evaluate L-1[ s/(s2+1)2 ]
--- Content provided by FirstRanker.com ---
(b) Obtain Fourier series for the function f (x) =x—x2 over—p < x < p and hence show that ?n=18 1/n2 = p2/12. 07
Attempt any one of the following. 07
- Obtain the Fourier series for the function f(x)=x2, -p < x < p. Hence show that ?n=18 (-1)n+1/n2 = p2/12.
- Find the Fourier series to represent the function f(x) given by f(x) = x for 0<x<p = 2p-x for p<x<2p. Hence show that ?n=18 1/(2n-1)2 = p2/8
- Obtain half-range cosine series for the function f(x) = x for 0<x< p/2 = p-x for p/2<x<p.
--- Content provided by FirstRanker.com ---
OR
Expand f(x)=eax as a Fourier series in the interval (~l,l). 07
(b) Attempt any two of the following. 07
- Express the following function as Fourier integral f (x) = 1 for |x|<1 = 0 for |x|>1 Hence evaluate, a) ?08 sinw cosx/w dw and b) ?08 Sln x/X dx.
- Show that d/dx {xnJn (x)} =xnJn-1(x)
- Show that (2n +1)xPn(x)=(n+1)Pn+1(x)+nPn-1(x).
--- Content provided by FirstRanker.com ---
Q.5 (a) Solve the one-dimensional wave equation together with following initial & boundary conditions. 07
?2u/?t2 = c2 ?2u/?x2 . Where, c2 = T/m.
u(0,t)=u(l,t)=0, ?t>0
u(x,0)= f(x) and ut(x,0)= g(x), ?0 < x < l
--- Content provided by FirstRanker.com ---
(b) Attempt any two of the following. 07
- xp+yg=3z
- (mz—ny)p+(nx—lz)q = (ly —mx)
- (x2 -y2 -z2)p +2xyq=2xz
--- Content provided by FirstRanker.com ---
Q.5 (a) A homogenous rod of conducting material of length 100 cm. has its ends kept at zero temperature and the temperature initially is 07
u(x,0) = x for 0<x<50
100 —x for 50 <x <100
Find the temperature u(x,t) at any time t, at a distance x.
(b) Attempt any two of the following. 07
--- Content provided by FirstRanker.com ---
- p-q=x-y
- z=p2+q2
--- Content provided by FirstRanker.com ---
This download link is referred from the post: GTU BE 2019 Summer Question Papers || Gujarat Technological University