Integral Transform-I - BU 2021 Question Paper

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EF-467

M.A/M.Sc. III^rd Semester (Reg./Pvt./ATKT)

Examination, 2021-22

Mathematics

Paper - X

Integral transform-I

Time: 3 Hours] [Maximum Marks: Reg.=85 Pvt.=100

Note:- Attempt all the questions.

SECTION - 'A'

Objective Type Questions

10x1.5=15

1. Choose the correct answer:

(i) L {e^at} is equal to

(a)

(b)

(c)

(d)

(ii) L^-1

is equal to -

(a) Cosat

(b) Coshat

(d) Sinht

(iii) L {y (t)} is equal to

(a) &bar;y(t)

(b) &bar;y(s)

(iv) In integral equation of the form

is called.

(a) Exact equation

(b) Convolution type

(d) Abel's equation

(v) The equation for one dimensional flow of heat along a bar is in the form

(a)

(b)

(c)

(d) None of these

(vi) One dimensional wave equation is given by

(a)

(b)

(c)

(d)

(vii) Equation of electrical circuit if condenser is not present is given by -

(a)

(b)

(c)

(d)

(viii) In applications of Beams y=y'=0 denotes -

(a) Free end

(b) Clamped end

(d) None of these

(ix) An equation of the type

is called -

(a) Two dimensional heat equation

(b) Heat equation in a cylindrical solid

(d) One dimensional heat equation

(x) L

is given by

(a)

(b)

(c)

(d) None of these

SECTION - 'B'

Short Answer Type Questions

5x5=25

2. Find

Find

3. Solve the differential equation T''(t) + Y'(t) + rY(t) = 0 under the conditions that Y(0)=1 and Y(t) and its derivatives have transforms.

Solve

4. Find the solution of two dimensional laplace's equation in cartesian coordinates by the method of separation of variables.

A tightly stretched string fixed end points x = 0 and x = l is intially in a position given by y (x,0) = y_o sin (πx/l). If it is released from rest from this position, find displacement y at any distance x from one end at any time t.

5. An alternating E.M.F. Esinωt is applied to an inductance and a capacitance C in series. Show that the current in the circuit is.

A beam which is hinged at its ends x = 0 and x = l carries a uniform load w_o per unit length. Find the deflection at any point.

6. Drive the equation for one dimensional flow of heat along a bar.

Find the solution of the equation

which tends to zero as x → ∞ and which satisfies the conditions -

u = f(t) when x = 0, t > 0
u = 0 when x > 0, t = 0

SECTION - 'C'

Long Answer Type Questions

9x5=45

7. If

, then prove that

Apply the convolution theorem to show that

8. Using laplace transform, find the solution of y'' + 25y = 10 cos 5t, where y (0) = 2, y'(0) = 0

Solve (D - 2) x - (D + 1) y = 6e^3t

(2D - 3) x + (D - 3) y = 6e^3t

x (0) = 3, y (0) = 0

9. Suppose that a uniform bar insulated over the sides has a tempreture distribution u = f(x) initially. Its ends foces x = 0 and x = 1 are suddenly coot ceoled at 0°C and miantained at that tempreture. Determine the subsequent tempreture at a subsequent time t

An infinite long string having one end at x = 0 is initially at rest on the x - axis. The end x = 0 undergoes a periodic transverse displacement given by A_o Sin nt, t > 0. Find the displacement of any point on the string at t > 0.

10. An inductor of 3 henrys is in series with a resistance of 30 ohms and an e.m.f. of 150 volts. Assuming that t = 0 the cur rent is zero, find the current at time t > 0.

A beam which is clamped at its ends x = 0, x = l carries a uniform load W_o per unit length. Show that the deflection at any point is

11. Determine the solution of one dimensional heat equation

under the boundary conditions θ (0,t) = 0.

θ (l,t) = 0, t > 0 and the initial condition θ (x,0) = x, 0 < x < l

l being the length of the bar.