Download GTU BE/B.Tech 2018 Winter 3rd Sem Old 130001 Mathematics Iii Question Paper

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Seat No.: Enrolment No.

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER-III (OLD) EXAMINATION - WINTER 2018

Subject Code:130001 Date:17/11/2018

Subject Name:Mathematics-I11

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Time:10:30 AM TO 01:30 PM

Instructions:

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

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Q1 (a) (1) Solve x(x—l)? —(x-2)=x>(2x-1) 03

X

(2) Solve (x*y% +2)ydx + 2—xy*)xdy =0 04

(b) Find the power series solution of the equation 4x² d²y/dx² + 2x dy/dx +y =0 about x =0. 07

Q2 (a) (1) Solve(D² —4)y =x² + sin2x. 03

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2

(2) Solve d²y/dx² — dy/dx — 2y= x/(1+e^x) 04

(b) Solve x²d²y/dx² +4x dy/dx +2y=e^x by using method of variation of parameter. 07

OR

Q3 (a) Find series solution of the differential equation (1 + x²) y'' +xy'—y=0. 07

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(b) (1) Show that ?₀¹ [x⁴ (1 - x)⁴dx = 1/630 03

(2) Prove that J_n(xⁿJ_n (x)) = xⁿ J_n-1(x) 04

Q3 (a) Find the half range cosine series for f(x) = { kx ; 0<x<1/2 k(l-x) ;1/2<x<1 07

Also Prove that S_n=1⁸ 1/(2n-1)² =p²/8

OR

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(b) Find the Fourier series of f(x) =xsin x in the interval (—p, p ) . Hence, deduce that S_n=1⁸ -1/(4n²-1) = (p-2)/4

Q4 (a) (1) Evaluate ?₀¹ x⁵ (log x)⁵ dx 03

(2) Find half range Fourier sine series of the function f(x)=p —x for 0<x< p. 04

(b) (1) Find the Laplace transform of the function f(t) = sin vt. 03

Total Marks: 70

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(2) Find the inverse Laplace transform of the function F(s) = (s+3)/(s² —4s+20) 4

(b) Solve the differential equation using Laplace Transformation method d²y/dt² +y =tcost, Given that y(0) =0, y'(0)=0, t > 0. 07

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OR

Q4 (a) (1) Find the Laplace transform of the function f(t) = t cost 03

(2) Find the inverse Laplace transform of the function F(s) =log[1 + (1/s²)] 04

(b) Define Convolution theorem for Laplace transform. Using Convolution theorem to find Laplace inverse of the function F(s) = 1/((s² +a²)(s² +b²)) 07

Q5 (a) (1) Form the partial differential equation of f (xy +z², x + y+ z) =0. 03

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(2) Solve(y +z)p+(x+z)q=x+y. 04

(b) Solve by the method of separation of variables ?²u/?x² = ?u/?y + 2u, u(0,y) =0 07

OR

Q5 (a) Using method of separation of variables, solve ?u/?x + ?u/?y = 3u, given that u(0,y) =3e^-2y —e^-5y. 07

(b) (1) Solve (x² +y² +1)dx —2xydy =0 03

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(2) Solve (D³ +3D² + 2D)y = x² +4x +8by using method of undetermined coefficients. 04

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