Download AKTU B-Tech 4th Sem 2016-17 RAS203 Mathematics Ii Question Paper

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B.TECH.

THEORY EXAMINATION (SEM–IV) 2016-17

MATHEMATICS-II

Time: 3 Hours

Max. Marks : 70

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Note: Be precise in your answer. In case of numerical problem assume data wherever not provided.

SECTION – A

1. Attempt any seven parts for the following: 7 x 2 = 14

1. Solve the differential equation d²y/dx² = -12x² + 24x – 20 with the condition x = 0, y = 5 and x = 0, dy/dx = 21 and hence find the value of y at x = 1.
2. For a differential equation d²y/dx² + 2ady/dx + y = 0, find the value of a for which the differential equation characteristic equation has equal roots.

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3. For a Legendre polynomial prove that Pn (1) = 1 and Pn (-1) = (-1)^
4. For the Bessel's function Jn(x) prove the following identities: J-n(x) = (-1)^Jn(x) and J-n(-x) = (-1)^Jn(x)
5. Evaluate the Laplace transform of Integral of a function L{?f(t)dt}.
6. Evaluate the value of integral ?08 t. e?²?cost dt.
7. Find the Fourier coefficient for the function f(x) = x² 0<x<2p

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8. Find the partial differential equation of all sphere whose centre lie on Z-axis.
9. Formulate the PDE by eliminating the arbitrary function from f(x² + y², y² + z²) = 0
10. Specify with suitable example the clarification Partial Differential Equation (PDE) for elliptic, parabolic and hyperbolic differential equation.

2. Attempt any three parts of the following questions: 3 x 7 = 21

1. A function n(x) satisfies the differential equation d²n(x)/dx² - n(x)/L² = 0, where L is a constant. The boundary conditions are n(0) = x and n(8) = 0. Find the solution to this equation.

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2. Find the series solution by Frobenius method for the differential equation (1 - x²)y" – 2xy' + 20y = 0
3. Determine the response of damped mass – spring system under a square wave given by the differential equation y" + 3y' + 2y = u(t - 1) - u(t – 2), y(0) = 0, y'(0) = 0 Using the Laplace transform.
4. Obtain the Fourier expansion of f (x) = x sin x as cosine series in (0, p) and hence show that 1/(1×3) + 1/(3×5) + 1/(5×7) + ... = p/4 - 1/2
5. Solve by method of separation of variable for PDE x ?u/?x + 20 ?u/?y = 0, u(x, 0) = 4e?x

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3. Attempt all parts of the following questions: 7 x 5 = 35

Attempt any two parts of the following:

1. Find the particular solution of the differential equation d²y/dx² + a²y = sec ax

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2. If y = y1(x) and y = y2(x) are two solutions of the equation d²y/dx² + P(x)dy/dx + Q(x)y = 0, then show that y1 (dy2/dx) - y2 (dy1/dx) = ce?Pdx, where c is constant.
3. Solve by method of variation of Parameter for the differential equation : d²y/dx² - 3dy/dx + 2y = e^(3x) / (1+e^x)

4. Attempt any two parts of the following:

1. Prove that J3/2(x) = v(2/px) (sinx/x - cos x)
2. Show that Legendre polynomials are orthogonal on the interval [-1,1]

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3. Prove that ?1¹ xP?(x)dx = 2n/(4n²-1)

5. Attempt any two parts of the following:

1. Find the Laplace transform of SRW – tooth wave function F(t) = Kt in 0<t<1 with period 1
2. Use Convolution theorem to find the inverse of function F(s) = 1/(s²+2s+5)
3. Solve the simultaneous differential equation, using Laplace transformation – dx/dt + 2x = sin 2t; dy/dt + 2y = cos 2t, where x (0) = 1, y (0) = 0

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6. Attempt any two parts of the following:

1. If f(x) = x², 0 < x < 2p then show that f(x) = p²/3 + 4 S(n=1 to 8) (cos nx)/n², where symbols have their usual meaning.
2. Find the complete solution of PDE (D² + 7DD' + 12D'²)z = e^(5x + tan(y - 3x))

7. Attempt any one part of the following:

1. A square plate is bounded by lines x = 0, y = 0; x = 20, y = 20. Its faces are insulated. The temperature along the upper horizontal edge is given by u (x, 20) = x (20 - x) when 0 < x < 20 while the upper three edges are kept at 0°C. Find the steady state temperature.

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2. A bar of 10 cm long with insulated sides A and B are kept at 20°C and 40°C respectively until steady state conditions prevail. The temperature at A is then suddenly varies to 50°C and the same instant that at B lowered to 10°C. Find the subsequent temperature at any point of the bar at any time.