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MPH 1: Mathematical Methods in Physics

Title Name	IGNOU MPH 1 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCPH
Course Name	Master of Science (Physics)
Subject Code	MPH 1
Subject Name	Mathematical Methods in Physics
Year	2026
Session	-
Language	English Medium
Assignment Code	MPH 1/Assignment-1/2026
Product Description	Assignment of MSCPH (Master of Science (Physics)) 2026. Latest MPH 001 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Format	Ready-to-Print PDF (.soft copy)

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January 2026 Session: 31st March, 2026
July 2026 Session: 30th September, 2026

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MPH 1 2025 - English

Tutor Marked Assignment

MATHEMATICAL METHODS IN PHYSICS

Course Code: MPH-001

Assignment Code: MPH-001/TMA/2025

Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

PART A

a) The 1-D wave equation for e.m. wave propagation in free space in given by (for $vec{E}parallelhat{y}$ )

$frac{partial^2 E_y}{partial x^2}-frac{1}{c^2}frac{partial^2 E_y}{partial t^2}=0$

Solve this equation if E_y = 0 at x = 0 and x = L.

b) The Helmholtz equation in Cartesian coordinates can be written as

$left(frac{partial^2}{partial x^2}+frac{partial^2}{partial y^2}+frac{partial^2}{partial z^2} ight)f(x,y,z)+kappa^2 f(x,y,z)=0$

Reduce it to three ODEs.

c) Using the generating function for Bessel functions of the first kind and integral order, obtain the recurrence relation

$J_{n-1}(x)+J_{n+1}(x)=2frac{n}{x}J_n(x)$

d) Using the Rodrigue's formula of Legendre polynomials, obtain the values of P₃(x) and P₄(x).

e) Write an expression for generating function for Hermite polynomials. Using this expression evaluate the integral

$int_{-infty}^{+infty}xe^{-x^2}H_n(x)H_m(x),dx$

2. a) What are orthonormal vectors? Show that two non-null, orthogonal vectors are linearly independent.

b) Show that the set of all 2x2 Hermitian matrices form a four dimensional real vector space. Obtain a suitable basis for this vector space.

c) Obtain the eigenvalues and the orthonormal eigenvectors for the following real symmetric matrix:

$egin{bmatrix}cos heta&sin heta&0\sin heta&-cos heta&0\0&0&-1end{bmatrix}$

d) Define contravariant and covariant tensors of rank 2. Prove that vⁱ = g^ijaj transform contravariantly.

PART B

3. a) Using the method of Residues, prove that

$int_{0}^{pi}frac{d heta}{1+sin^2 heta}=frac{pi}{sqrt{2}}$

b) Show that the series $sum_{n=1}^{infty}frac{z^{n-1}}{2^n}$ converges for IzI < 2 and find its sum.

c) Write the Laurent series expansion of ez/ (z-1)² singularity and the region of convergence. about z = 1. Determine the type of

d) Obtain the images of the line x = 0 and y = 0 under the transformation w = z² and prove that they intersect at right angles.

4. a) Obtain the Fourier cosine transform of the function

$f(x)=e^{-lambda x},quad 0<x<infty,quadlambda>0$

b) Solve the initial value problem using the method of Laplace transform

$y''-2y'-3y=0,quad y(0)=1,quad y'(0)=7$

c) A metal plate covering the first quadrant of the xy plane has its edge along y-axis insulated. The edge along x-axis is held at an initial temperature

$T(x,0)=egin{cases}100(2-x)& ext{for}0<x<2\0& ext{for}x>2end{cases}$

Obtain the steady state temperature distribution as a function of x and y.

d) Determine the inverse Laplace transform of

$F(x)=frac{x+1}{x^3+x^2-6x}$

5. a) Construct the multiplication table for the group of permutations of {1,2,3}.

b) Obtain all the proper subgroups of the permutation groups S₃ = {e,p1...,p5 }and their cosets.

Given:

Image ignou-ignouacademy-com-ignou-mph-1-solved-assignment-html-p-ignou-94250

c) Show that in the x - y plane, a rotation about the origin in an anti-clockwise

direction by an angle φ will move the points (x, y) to by the matrix

$egin{pmatrix}cosphi&-sinphi\sinphi&cosphiend{pmatrix}$

MPH 001 (January 2026 - July 2026) - ENGLISH

Tutor Marked Assignment

MATHEMATICAL METHODS IN PHYSICS

Course Code: MPH-001

Assignment Code: MPH-001/TMA/2026

Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

PART A

1. a) Reduce the following PDE into three ODEs:

$$\left[ \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right] f(x, y, z) + k^2 f(x, y, z) = 0$

b) Derive an integral equation corresponding to the ODE:

$$y'' - 2y = 0$
subject to the conditions: $y(0) = 4; y'(0) = -2$

c) Use the method of separation of variables to reduce the Laplace's equation $\nabla^2 f = 0$ into three ODEs.

d) Using the generating function for Bessel functions of the first kind and integral order

$$g(x, t) = \exp \left[ \frac{x}{2} \left( t - \frac{1}{t} \right) \right] = \sum_{n=-\infty}^{\infty} J_n(x) t^n$

Obtain the recurrence relation

$$J_{n-1}(x) + J_{n+1}(x) = 2 \frac{n}{x} J_n(x)$

Also using the generating function show that

$$J_0(x) + 2J_2(x) + 2J_4(x) + 2J_6(x) + \dots + 2J_{2k}(x) + \dots = 1$

2. a) Obtain orthogonality relation for Hermite polynomials using the generating function:

$$e^{2xt-t^2} = \sum_{n=0}^{\infty} H_n(x) \frac{t^n}{n!}$

b) i) Show that the following vectors

$$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}$

are linearly independent.

ii) The first Pauli matrix is

$$\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
calculate $U_1(\theta) = \exp(i\theta \sigma_1) = 1 + i\theta \sigma_1 - \frac{\theta^2}{2} \sigma_1^2 - \dots$

For real $\theta$ , show that U₁ is unitary and has determinant 1.

c) Obtain the eigenvalues and eigenvectors of matrix A:

$$A = \begin{bmatrix} 2 & 0 & -1 \\ 0 & 2 & 0 \\ -1 & 0 & 2 \end{bmatrix}$
d) Define covariant and cotravariant tensors of rank 2. Prove that $a_i = g_{ij}v^j$ transform covariantly, where g_ij are the components of the matrix tensor of rank 2 and vⁱ the components of a contravariant vector.

PART B

3. a) i) Obtain the analytic function whose real part is

$$u(x, y) = e^x \cos y$

ii) Locate and name the singularity of the function:

$$\frac{z^2 - 2z}{z^2 + 2z + 2}$
b) Calculate the value of the integral $\oint_C \frac{\cos z}{z} dz$ when C is the circle $|z| = 2$ .
c) Show that the Series $\sum_{n=1}^{\infty} z^n (1 - z)$ converges for |z| < 1 and find its sum.
d) Obtain the Laurent series expansion of $\frac{e^z}{(z - 1)^2}$ about $z = 1$ . Determine the type of singularity and the region of convergence.
e) Evaluate the value of the contour integral $6 \oint_C \frac{z \, dz}{(z - 1)(z^2 + 9)}$ where C is a circle defined by $|z| = 4$ .

4. a) Evaluate the integral

$$\int_0^{2\pi} \frac{d\theta}{1 + p \cos \theta}$

by the method of residues when -1 < p < 1.

b) Consider a triangle T in the z-plane with vertices at i, 1 - i, 1 + i. Determine the triangle T₀ into which T is mapped under the transformations

$$w = iz + 2 - i$

c) Obtain the Fourier cosine transformation of the function:

$$f(x) = e^{-px}, \quad 0 < x < \infty, \quad p > 0$

d) Define homomorphisms. When do the homomorphisms become endomorphism and isomorphism?

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