IGNOU PHE 5 SOLVED ASSIGNMENT 2025

Ques 1.

Solve the following ordinary differential equations:

i) 

ii) 

iii) 

Ques 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?

Ques 3.

A block of mass 1 kg is attached to a spring of force constant  .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. 

Ques 4.

Solve the following ODE using the power series method:

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

Ques 5.

Show that the function

i) u(x,y)=  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.

Ques 6.

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

f (x, y) = sin xy

Ques 7.

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

Ques 8.

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

Ques 9.

Obtain a solution of the wave equation

Ques 10.

निम्नलिखित साधारण अवकल समीकरणों का हल करें:

i) 

ii) 

iii) 

Ques 11.

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।

Ques 12.

1 kg द्रव्यमान के एक पिंड को एक कमानी से जोड़ा जाता है जिसका बल गुणांक 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।

Ques 13.

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

Ques 14.

दिखाएं कि फलन

i)  द्विविम लाप्लास समीकरण का हल है।

ii) 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.

Ques 15.

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

Ques 16.

आवर्तकाल  वाले निम्नलिखित आवर्ती फलन का फूरिए श्रेणी विस्तार प्राप्त करें :

  के लिए

Ques 17.

निम्नलिखित परिसीमा और आदि मान प्रतिबंधों :

u(0,1) = u(π,1) = 0,

u(x,0)=x(n-x) और 

के लिए, तरंग समीकरण

का हल प्राप्त करें।

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