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

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GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER-III (Old) EXAMINATION - WINTER 2019

Subject Code: 130001 Date: 22/11/2019

Subject Name: Mathematics-III

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

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Instructions:

1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

1. 1. Q.1 Obtain Fourier series to represent f(x) = (p – x) in the interval 0 < x < 2p. 07

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   2. Q.2 Using Laplace Transform solve the given IVP. y''+y=t y(0)=1 y'(0)=1. 07
   3. Q.3 Solve by Method of Variation of Parameters. y''+4y = sec²x. 07
   4. Q.3 Solve 3(3x+2)²y"+3(3x+2)y'-36y =3x² +4x+1. 07
2. 1. OR Obtain series solution of y'''-xy =0. 07

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   2. (1) Solve linear differential equation x dy/dx +2xy = 2. 03
   3. (2) Solve dy/dx + y/x log x= y²/x (log x)². 04
3. 1. Q4 Using method of undermined co-efficient method to solve y''+3y' +10y =25x² +3. 07

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4. 1. OR
   2. (1) Solve (x²y² +xy+1)ydx+(x²y² —xy+1)xdy=0. 03
   3. (2) Solve (1+ y²)dx+(x—e^{tan-1 y})dy =0. 04
5. 1. Q4 Using method of undermined co-efficient method to solve y''-9y = x+e^x —sin 2x. 07

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6. 1. Q.5 (1) Find the inverse Laplace Transform of (5s² +3s-16)/((s-1)(s-2)(s-3)). 03
   2. (2) Using convolution theorem, find the inverse Laplace transform of (s+2)/(s² +4s+5). 04

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1. 1. Q.5 (1) Find the Fourier Transform of the function f(x) = e^-|x|. 04
   2. (2) Evaluate ?₀⁸ x²e^-x2dx. 03
2. 1. (1) By using first shifting theorem, obtain the value of L{(t+1)²e^t}. 03

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   2. (2) Find L{log(s+a)/(s+b)}. 04
3. 1. (1)Find the Fourier Cosine transform of the function f(x) = k, 0<x<a and 0, x>a. 04
   2. (2) Evaluate ?₀⁸ (sin t/t)²dt. 03

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4. 1. Prove that (1) J_-1/2(x) = v(2/(px)) cos x
   2. (2) J_1/2(x) = v(2/(px)) sin x 07
5. 1. A taut string of length l has its ends x=0 and x=l fixed. The mid-point is stretched to a small height and released from rest at time t=0. Find the displacement u(x,t). 07

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6. 1. OR Show that ?_-1¹ P_m(x)P_n(x)dx=0, if m?n and ?_-1¹ [P_n(x)]²dx= 2/(2n+1), if m = n. (m, n being integers) 07
   2. Solve u_xx +u_yy =0 which satisfies the boundary condition u(0,y)=u(a,y)=0 for 0 <y <b and u(x,b)=0, u(x,0)= f(x) for 0<x<a. 07