Download DU (University of Delhi) B-Tech 5th Semester 6310 Mathematical Physics II Question Paper

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SI. No. of Ques. Paper 16310 F-5

Unique Paper Code 12341504

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Name of Paper : Mathematical Physics - II

Name of Course : B. Tech. (Computer Science) (FYUP Scheme)

Semester

Duration : 3 hours Maximum Marks : 75

(Write your Roll No. on the top immediately on receipt of this question paper.)

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Do five questions in all. Question No. 1 is compulsory.

1. Do any five questions :

(a) Determine the order, degree and linearity of the differential equation:

d²y / dx² + (dy / dx) + y = 0

(b) What is Wronskian? Calculate the value of wronskian for:

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x and xⁿ(lnx)

(c) Prove the following property of Poisson Bracket:

[u, [v, w]] = [[u,v], w] + [v, [u, w]]

(d) Find the extreme points of the function:

F(x,y) = y + 4xy + 3x² + x³

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(e) Solve:

dy/dx + y/x = xⁿ

(f) Define generalised momenta for n-particle system, and find its time derivative.

(g) Form the differential equation whose only solutions are:

a₁, a₂, a₃ & a₄

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(h) Find the extremal of the integral :

∫(2y sinx - y'²)dx, here y' = dy/dx

from 0 to π/2

2. (a) Solve the following differential equation:

dy/dx = y tanx - y² secx

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(b) dy/dx = (x-y-2) / (x+y+5)

3. Solve the following differential equations:

(a) d²y/dx² - y = x cosx

(b) x² d²y/dx² + 3x dy/dx - y = x^m

4. (a) Solve the following differential equation:

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(x² + y²)dx - xy²dy = 0

(b) Using the method of variation of parameters, solve

(D² + 9)y = x sin(3x) where D = d/dx

5. (a) Using the method of undetermined coefficients, solve

d²y/dx² - 2 dy/dx - y = e^x + x

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(b) Solve the coupled differential equations:

dx/dt + 2x = dy/dt + 10 cost

dy/dt + 2y = dx/dt

6. (a) Find the equation of the shortest path between two points on the surface of sphere of radius a.

(b) Using Lagrange’s method of undetermined multiplier, find the maximum value of x^p y^q z^r when the variables x, y, z are subjected to the condition ax + by + cz = p + q + r.

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7. (a) Find the Hamiltonian for

L = (1/2)a q̇₁² + (1/2)b q̇₂² - V(q₁, q₂)

(b) Using Hamilton’s equations of motion and the expression

L(q, q̇) = p q̇ - H(q,p)

prove that: ṗ = -∂H/∂q and ṗ = ∂L/∂q̇

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8. (a) Show that

(i) [q_j, H] = ∂q_j

(ii) [p_j, H] = ∂p_j

here, H denotes Hamiltonian and 1 ≤ j ≤ n.

(b) Write the Lagrangian of the system of two masses 2m and m, shown below in Fig. (1). In this figure, y₁ and y₂ are the displacements of two masses from their equilibrium positions. Hence obtain the equations of motion of these two masses.

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(Figure 1 is missing, so I can't include it)

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