Download GTU BE/B.Tech 2019 Summer 3rd Sem Old 130002 Advanced Engineering Mathematics Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY

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

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Subject Code: 130002 Date: 30/05/2019

Subject Name: Advanced Engineering Mathematics

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.

Q.1

(a) (i) Solve 3e^xtanydx + (1 — e^x) sec²ydy=0

(i) Solve y' - (1 +3x)y=x+2; y(1)=e -1

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(b) Find the Power series solution of the differential equation y" =y

Q.2

(a) Using the method of separation of variables solve u_xx= 16 u_y.

(b) Find the series solution of the differential equation by Frobenius method

d²y/dx² + dy/dx = 0

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OR

(b) (i) Solve y" + 4y =8 cos2x, y(0) =0, y'(0) =2

(ii) Solve y"- 4y' - 12y = 7 e^7x by method of undetermined coefficients.

Q.3

(a) Find the Fourier series for the function f(x) = x²+ x, -1 < x < 1.

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(b) Find the Fourier series of the function

f(x) = { -p, 0 < x < p ; x—p, p < x < 2p }

OR

Q.3 (a) Find the Fourier series with period 3 to represent f(x) = 2x — x² in the range (0, 3).

(b) Find the half range Fourier cosine series of the function f(x) = c — x in interval (0, c) with period 2c.

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Q.4

(a) (i) Find the Laplace transform of e^-t (4t² + 3cos2t + 2e^-2t)

(ii) Prove that L(sinat) = a/(s²+a²) and L(cosat) = s/(s²+a²), s > 0, where a is a constant.

(b) Find the Inverse Laplace transform of

(i) (s+3)/((s²+1)(s²+9)) (ii) (2s+3)/(s²—2s+5)

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OR

Q.4 (a) (i) Find the Laplace transform of ?₀^t e^tt cost dt

(ii) Find the Inverse Laplace transform of (1+ e^-ps/2)/(s²+4)

Q.5

(a) (i) Form Partial differential equation by eliminating the arbitrary function from the equation z=y²+ 2f(1/x + logy)

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(ii) Define the following: (1) Beta function (2) Dirac’s Delta Function

(b) Express the function as a Fourier Integral

f(x) = { 1, |x| < 1 ; 0, |x| > 1 }

OR

Q.5 (a) (i) Solve: p+q= pq

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(ii) Solve: x (y² — z²) p + y (z² — x²) q = z(x² — y²).

(b) Solve the following:

(i) ?³z/?x³ - 2(?³z/?x²?y) = sin (3x + 2y)

(ii) (D-D'-1)(D-D'-2)z=e^x+y

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