Download GTU B.Tech 2020 Summer 3rd Sem 130001 Mathematics Iii Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER- III EXAMINATION - SUMMER 2020

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Subject Code: 130001 Date:26/10/2020

Subject Name: MATHEMATICS-III

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) (i) Solve y' + x - 2y = e^-x 03

(ii) Solve (x³ + 3xy²)dx + (3x²y + y³)dy = 0 04

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(b) Find the power series solution of the equation x²y'' + xy = 0. 07

Q2

(a) (i) Solve y'' - 5y' + 6y = e^x 03

(ii) Using the method of variation of parameter, solve y'' - 4y' + 4y = e^2x 04

(b) Using the method of undetermined coefficient, solve y''' + 3y'' + 2y' = x² + 4x + 8 07

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OR

(b) Solve the equation by series method (x-2)y''' - x²y' + 9y = 0 about x = 0. 07

Q.3

(a) Find the Fourier series of f(x) = x + x² in the interval (-p, p). Hence, deduce that

p²/6 = 1 + 1/2² + 1/3² + ... 07

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(b) Find the Fourier series of f(x) = e^x, -p < x < p. 07

OR

Q.3

(a) Find the Fourier series of f(x) = |x|, -p < x < p. 07

(b) Find the Half range Fourier cosine series of f(x) = xsinx, 0 < x < p and f(x+2p) = f(x) 07

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Q4

(a) (1) Find the Laplace transform of the function f(t) = e^t sin 2t. 03

(2) Find the inverse Laplace transform of the function F(s) = (2s-5) / (s² + 8s + 25). 04

(b) Solve the differential equation using Laplace Transformation method y'' - 2y' + y = sint, Given that y(0) = 1, y'(0) = 0, t > 0. 07

OR

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Q4

(a) (1) Find the Laplace transform of the function f(t) = t cos²t 03

(2) Find the inverse Laplace transform of the function F(s) = (6s - 4) / (s² - 4s + 20) 04

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

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Q.5

(a) (i) Form the partial differential equation of f(x² - y, xy + z) = 0. 03

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

(b) Solve by the method of separation of variables ?²u / ?x² - 5?u/?x + 6?u/?y = 0 07

OR

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Q.5

(a) (i) Solve p² + q² = x + y 03

(ii) Solve pyz - zxq = xy 04

(b) Find the Fourier integral of the function f(x) = 1, |x| < 1 and f(x) = 0, |x| > 1. Hence, evaluate (i) ?(Tsin?cos?x)/? d? (ii) ?(Tsin?)/? d? 07

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