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PHE 5: Mathematical Methods in Physics-II

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Title Name	IGNOU PHE 5 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	PHE 5
Subject Name	Mathematical Methods in Physics-II
Year	2026
Session	-
Language	English Medium
Assignment Code	PHE 5/Assignment-1/2026
Product Description	Assignment of BSC (Bachelor in Science) 2026. Latest PHE 05 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).
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PHE 5 2025 - English

Tutor Marked Assignment

MATHEMATICAL METHODS IN PHYSICS-II

Course Code: PHE-05

Assignment Code: PHE-05/TMA/2025

Max. Marks: 100

Note: Attempt all questions. Symbols have their usual meanings. The marks for each question are indicated against it.

1. a) Solve the following ordinary differential equations:

i) $frac{dy}{dx}=frac{y-x}{x-4y}$

ii) $left(2yx^{2}+4 ight)frac{dy}{dx}+left(2y^{2}-3 ight)=0$

iii) $frac{d^{2}y}{dx^{2}}+3frac{dy}{dx}-10y=3x^{2}$

2. Newton’s Law of Cooling states that the rate at which the object's temperature decreases is directly proportional to the difference between the temperature of the object and the ambient temperature. A cup of tea (temperature = 90°C) is placed in a room whose temperature is 30°C. After five minutes, the temperature of the tea has dropped to 70°C. After how much time would the temperature of the tea be 50°C?

3. A block of mass 1 kg is attached to a spring of force constant $k=frac{25}{4}Nm^{-1}$ .It is pulled 0.3 m from its equilibrium position and released from rest. This spring‐block apparatus is submerged in a viscous fluid medium which exerts a damping force of – 4 v (where v is the instantaneous velocity of the block). Determine of the position x(t) of the block at time t.

4. Solve the following ODE using the power series method:

(x+2)y"+xy'-y=0

5. a) Show that the function

i) u(x,y)= $tan^{-1}left(frac{y}{x} ight)$ is a solution of the two-dimensional Laplace’s equation.

ii) u(x,t)=e^{-6x cos2x}is a solution of the one-dimensional heat equation.

b) Obtain all the first and second order partial derivatives of the function:

f (x, y) = sin xy

6. Reduce the following PDE to a set of three ODEs by the method of separation of variables.

$frac{1}{r}frac{partial}{partial r}left(rfrac{partial V}{partial r} ight)+r^{frac{1}{2}}frac{partial^{2}V}{partialTheta^{2}}+frac{partial^{2}V}{partial z^{2}}+k^{2}V=0$

7. Obtain the Fourier series expansion for the following periodic function which has a period of π:

Image ignouassignments-ignouacademy-com-ignou-phe-5-solved-assignment-html-p-solved-62912

8. Obtain a solution of the wave equation

Image ignouassignments-ignouacademy-com-ignou-phe-5-solved-assignment-html-p-assignment-31399

PHE 05 (January 2026 - July 2026) - ENGLISH

Tutor Marked Assignment PHE-05: Mathematical Methods in Physics-II

Course Code: PHE-05

Assignment Code: PHE-05/TMA/2026

Max. Marks: 100

Note: Attempt all questions. Symbols have their usual meanings. The marks for each question are indicated against it.

1. a) Solve the following ordinary differential equations:

(i) $x dx + y dy + 4y^3(x^2 + y^2)dy = 0$

(ii) $y'' + 4y = 2 \cos x \cos 3x$

b) Solve the initial value problem:

$\frac{d^2x}{dt^2} - 3\frac{dx}{dt} - 10x = 0, \quad x(0) = 1, \quad \dot{x}(0) = 0$

2. A parachutist is falling with a speed of $50 \text{ ms}^{-1}$ when his parachute opens. If the air resistance is (Mv²)/25 where M is the total mass of the man and his parachute, find the speed of the man as a function of time t after the parachute opens. Take $g = 10 \text{ ms}^{-1}$ .

3. A cylinder with a diameter $1.0 \text{ m}$ is stands in water with its axis vertical. When depressed slightly and released, it oscillates with a time period of $2 \text{ s}$ . Determine the mass of the cylinder.

4. Solve the following ODE using the Frobenius method:

$\frac{d^2y}{dx^2} + \frac{1}{2x} \frac{dy}{dx} + y = 0$

5. a) Show that:

$$u = e^{-\omega^2 c^2 t} \sin(\omega x)$
is a solution of the one-dimensional heat equation.

b) Determine all the first and second order partial derivatives for the function:

$$u(x, y) = x^2 \sin y + y^2 \cos x$

6. The surface of a ball of radius A is kept at a temperature zero. If the initial temperature in the ball is f(r), write down the boundary conditions and show that the temperature in the ball at time t, u(r, t), is the solution to the equation:

$$c^2 \left( \frac{\partial^2 u}{\partial r^2} + \frac{2}{r} \frac{\partial u}{\partial r} \right) = \frac{\partial u}{\partial t}$
7. Obtain the Fourier series for the following periodic function which has a period of $2\pi$ :

$$f(x) = \pi^2 - x^2 \text{ for } -\pi < x < \pi$

8. Determine the deflection u(x, t) of a vibrating string of length L, which has its ends fixed, corresponding to a zero initial velocity and an initial deflection given by the function:

$$f(x) = \begin{cases} Ax & 0 \le x \le L/4 \\ AL/4 & L/4 \le x \le 3L/4 \\ A(L - x) & 3L/4 \le x \le L \end{cases}$

PHE 5 2025 - Hindi

अध्यापक जांच सत्रीय कार्य भौतिकी में गणितीय विधियाँ-11

पाठ्यक्रम कोड: PHE-05

सत्रीय कार्य कोड: PHE-05/TMA/2025

अधिकतम अंक: 100

नोट: सभी प्रश्न हल करें। प्रतीकों के अपने सामान्य अर्थ हैं। प्रत्येक प्रश्न के अंक उसके सामने दर्शाए गए हैं।

1. (क) निम्नलिखित साधारण अवकल समीकरणों का हल करें:

i) $frac{dy}{dx}=frac{y-x}{x-4y}.$

ii) $left(2yx^{2}+4 ight)frac{dy}{dx}+left(2y^{2}x-3 ight)=0.$

iii) $frac{d^{2}y}{dx^{2}}+3frac{dy}{dx}-10y=3x^{2}.$

2. न्यूटन के शीतलन के नियम के कथन के अनुसार एक पिंड़ के तापमान के घटने का दर पिंड़ और परिवेश के तापमान के अंतर का अनुक्रमानुपाती है। एक चाय के प्याले (तापमान 90°C) को एक कमरे में रखा जाता है जिसका तापमान 30°C है। पांच मिनट के बाद चाय का तापमान घटकर 70°C हो जाता है। परिकलित करें कि कितने समय के बाद चाय का तापमान 50°C होगा।

3. 1 kg द्रव्यमान के एक पिंड को एक कमानी से जोड़ा जाता है जिसका बल गुणांक $k=frac{22}{4}$ Nm ^-1 है। पिंड को साम्यावस्था से 0.3m तक खींचकर, विरामावस्था से छोड़ा जाता है। इस कमानी-पिंड निकाय को एक श्यान तरल मध्यक में निमज्जित किया जाता है, जो -4 v का एक अवंमंदक बल लगाता है (जहां v पिंड का तात्क्षणिक वेग है)। समय t पर पिंड़ की स्थिति x(t) ज्ञात करें।

4. घट-श्रेणी विधि द्वारा निम्नलिखित सामान्य अवकल गुणांक का हल प्राप्त करें:

$(x+2)y''+xy'-y=0$

5. क) दिखाएं कि फलन

i) $u(x,y)= an^{-1}left(frac{y}{x} ight)$ द्विविम लाप्लास समीकरण का हल है।

ii) $u(x,t)=e^{-6t}cos 2x$ एकविम ऊष्मा समीकरण का हल है।

ख) निम्नलिखित फलन के सभी प्रथम और द्वितीय श्रेणी के आंशिक अवकलज प्राप्त करें :

f(x, y) = sin xy.

6. निम्नलिखित आंशिक अवकल समीकरण को चर पृथक्ककरण विधि द्वारा तीन साधारण अवकल समीकरणों के रूप में लिखें :

$frac{1}{r}frac{partial}{partial r}left(rfrac{partial V}{partial r} ight)+frac{1}{r^{2}}frac{partial^{2}V}{partial heta^{2}}+frac{partial^{2}V}{partial z^{2}}+k^{2}V=0$

7. आवर्तकाल $pi$ वाले निम्नलिखित आवर्ती फलन का फूरिए श्रेणी विस्तार प्राप्त करें :

$f(x)=egin{cases}frac{4}{pi}x,&0leq x<frac{pi}{2}\-frac{4}{pi}x,&-frac{pi}{2}leq xleq 0end{cases}$ के लिए

8. निम्नलिखित परिसीमा और आदि मान प्रतिबंधों :

u(0,1) = u(π,1) = 0,

u(x,0)=x(n-x) और $frac{partial u(x,0)}{partial x}=0$

के लिए, तरंग समीकरण

$frac{partial^2 u(x,t)}{partial t^2}=16frac{partial^2 u(x,t)}{partial x^2};quad 0leq xleqpi,t>0$

का हल प्राप्त करें।

PHE 05 (January 2026 - July 2026) - HINDI

अध्यापक जांच सत्रीय कार्य
पी. एच. ई. -05: भौतिकी में गणितीय विधियां -II

पाठ्यक्रम कोड : पी.एच.ई. -05
सत्रीय कार्य कोड : पी.एच.ई.-05/टी.एम.ए./ 2026
अधिकतम अंक : 100

नोट : सभी प्रश्न हल करें। प्रत्येक प्रश्न के अंक उसके सामने दिए गए हैं। प्रतीकों के अपने सामान्य अर्थ हैं।

1. क) निम्नलिखित साधारण अवकल समीकरणों को हल करें :

(i) $x dx + y dy + 4y^3(x^2 + y^2) dy = 0$

(ii) $y'' + 4y = 2 \cos x \cos 3x$

ख) निम्नलिखित आदि मान समस्या को हल करें :

$\frac{d^2x}{dt^2} - 3 \frac{dx}{dt} - 10x = 0, \quad x(0) = 1, \quad \dot{x}(0) = 0$

2. एक आदमी पैराशूट के साथ गिर रहा है। जब उसका पैराशूट खुलता है, उसकी चाल $50 \text{ ms}^{-1}$ है। यदि वायु प्रतिरोध (Mv²)/25 है, जहां M आदमी और पैराशूट का द्रव्यमान है, पैराशूट खुलने के बाद आदमी के चाल को समय t के फलन के रूप में प्राप्त करें। $g = 10 \text{ ms}^{-1}$ लें। (10) *(नोट: छवि में g की इकाई $\text{ms}^{-1}$ लिखी है, जो संभवतः $\text{ms}^{-2}$ होनी चाहिए)*

3. $1.0 \text{ m}$ व्यास वाला एक बेलन पानी में ऐसे खड़ा है कि उसका अक्ष ऊर्ध्वाधर दिशा में है। जब उसे थोड़ा सा दबाकर छोड़ दिया जाता है, वह $2 \text{ s}$ की आवर्तकाल के साथ दोलन करता है। बेलन का द्रव्यमान प्राप्त करें।

4. फोबेनियस विधि द्वारा निम्नलिखित साधारण अवकल समीकरण का हल प्राप्त करें :

$\frac{d^2y}{dx^2} + \frac{1}{2x} \frac{dy}{dx} + y = 0$

5. क) दिखायें कि फलन

$$u = e^{-\omega^2 c^2 t} \sin(\omega x)$

एक-विमिय ताप समीकरण का हल है।

ख) निम्नलिखित फलन के सभी प्रथम और द्वितीय कोटि के आंशिक अवकलज प्राप्त करें :

$$u(x, y) = x^2 \sin y + y^2 \cos x$

6. त्रिज्या A वाली एक गेंद की सतह को शून्य तापमान पर रखा जाता है। यदि गेंद में आदि तापमान f(r) है तो परिसीमा प्रतिबन्ध लिखें और दिखाएं कि समय t पर गेंद में तापमान u(r, t), निम्नलिखित समीकरण का हल है:

$$c^2 \left( \frac{\partial^2 u}{\partial r^2} + \frac{2}{r} \frac{\partial u}{\partial r} \right) = \frac{\partial u}{\partial t}$

7. $2\pi$ आवर्तकाल वाले निम्नलिखित आवर्ती फलन का फूरिए श्रेणी प्रसार प्राप्त करें :

$f(x) = \pi^2 - x^2, -\pi < x < 0$ के लिए।

8. लंबाई L वाले एक कम्पमान तार के दोनों सिरे स्थिर हैं। तार का आदि वेग शून्य है तथा आदि विक्षेपण निम्नलिखित फलन द्वारा परिभाषित है :

$$f(x) = Ax \qquad\qquad\qquad 0 \le x \le L/4$
$$\quad\;\;\; = AL/4 \qquad\qquad\quad L/4 \le x \le 3L/4$
$$\quad\;\;\; = A(L - x) \qquad\quad 3L/4 \le x \le L$

इन प्रतिबन्धों के लिए तार का विक्षेपण u(x, t) प्राप्त करें।

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