Download GTU BE/B.Tech 2019 Summer 3rd Sem Old 130001 Mathematics Iii Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER- ITI(OLD) EXAMINATION - SUMMER 2019

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Subject Code: 130001 Date:30/05/2019

Subject Name: Mathematics-I11

Time: 02:30 PM TO 05:30 PM Total Marks: 70

Instructions:

1. Attempt all questions.

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2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q1 (a) Obtain series solution of ^dy/_dx + y = 0. 07

(b) Attempt any two of the following. 07

1. ^dy/_dx = ^x(2logx+1)/_siny+ycosy

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2. xdy-ydx=v(x²+y²)dx
3. ^dy/_dx + ycotx = cosx
4. sin(y-xp)=p . Where, p = ^dy/_dx.

Q.2 (a) Obtain the Frobenius series solution of 2x²y" +3xy' — (x² + l)y =0. 07

(b) Attempt any two of the following. 07

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1. ^d4y/_dx⁴ +13 ^d2y/_dx²+36y=0
2. (D²+3D+2)y=5.
3. y" -2y' +y=cos2x
4. Solve by Method of variation of parameters. y" +a²y = tanax.

OR

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(b) Attempt any two of the following. 07

1. x^{2 d2y}/_dx² -x ^dy/_dx -By=x²log x.
2. Using method of undetermined multipliers solve y" +4y =8x².
3. Using method of undetermined multipliers solve y" —3y' +2y =e^x.
4. Prove that ?₀⁸ xⁿ e^-x dx = G(n+1)

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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

1. Find the Laplace transform of e^-3t +sin 2t + 5cosh²t.
2. Evaluate L(sin t sin 2t sin 3t).
3. Evaluate L[e^-t (cos4t +3sin 4t)].

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4. Evaluate L^-1{ ¹/_(s-1)(s²+1) }

OR

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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

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1. Evaluate L{ ^1-cos2t/_t }
2. Evaluate L(t³e^-3t)
3. Evaluate L^-1{ ¹/_s(s²+a²) }
4. 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 ]

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(b) Obtain Fourier series for the function f (x) =x—x² over—p < x < p and hence show that ?_n=1^{8 1}/_n² = ^p2/₁₂. 07

Attempt any one of the following. 07

1. Obtain the Fourier series for the function f(x)=x², -p < x < p. Hence show that ?_n=1⁸ (-1)ⁿ⁺¹/_n² = ^p2/₁₂.
2. 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=1^{8 1}/_(2n-1)² = ^p2/₈
3. Obtain half-range cosine series for the function f(x) = x for 0<x< p/2 = p-x for p/2<x<p.

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OR

Expand f(x)=e^ax as a Fourier series in the interval (~l,l). 07

(b) Attempt any two of the following. 07

1. Express the following function as Fourier integral f (x) = 1 for |x|<1 = 0 for |x|>1 Hence evaluate, a) ?₀^{8 sinw cosx}/_w dw and b) ?₀^{8 Sln x}/_X dx.
2. Show that d/dx {xⁿJ_n (x)} =xⁿJ_n-1(x)

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3. Show that (2n +1)xP_n(x)=(n+1)P_n+1(x)+nP_n-1(x).

Q.5 (a) Solve the one-dimensional wave equation together with following initial & boundary conditions. 07

^?2u/_?t² = c^{2 ?2u}/_?x² . Where, c² = ^T/_m.

u(0,t)=u(l,t)=0, ?t>0

u(x,0)= f(x) and u_t(x,0)= g(x), ?0 < x < l

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(b) Attempt any two of the following. 07

1. xp+yg=3z
2. (mz—ny)p+(nx—lz)q = (ly —mx)
3. (x² -y² -z²)p +2xyq=2xz

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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

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1. p-q=x-y
2. z=p²+q²

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