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BMTC 132: Differential Equations

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Title Name	IGNOU BSCFMT BMTC 132 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSCFMT
Course Name	Bachelor of Science (Mathematics)
Subject Code	BMTC 132
Subject Name	Differential Equations
Year	2026
Session	-
Language	English Medium
Assignment Code	BMTC 132/Assignment-1/2026
Product Description	Assignment of BSCFMT (Bachelor of Science (Mathematics)) 2026. Latest BMTC 132 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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BMTC 132 2025 - English

ASSIGNMENT

Course Code: BMTC-132

Assignment Code: BMTC-132/TMA/2025

Maximum Marks: 100

1. a) Solve the differential equations:

$frac{dy}{dx}=-frac{2x+y+1}{4x+2y-1}$

b) Solve the simultaneous equations:

$frac{dx}{y-z}=frac{dy}{z-x}=frac{dz}{x-y}$

2. a) By using the method of variation of parameter, find the general solution of the differential equation:

$y''+y=sec^2 x$

b) If:

Z= x²y+2xy²

where x = sin zt

and

y = cost

find dz/dt when t=0by (i) chain rule and by (ii) the direct substitution.

3. a) Verify that the differential equation:

z(x²-y²-z^²)dx+(x+z)xzdy

+ x(z²-x2-xy)dz = 0

is integrable and find its integral.

b) Solve:

$y'=frac{zy^4+x^4}{xy^3}$

4. a) Find the general solution of the equation:

x(y²-z²)u_x + y(z²-x²)u_y +z(x²- y²)u₂ = 0

b) Show that for the function f given by:

$f(x,y)=frac{xy}{x^2+y^2}$

$lim_{x ightarrow0}lim_{y ightarrow0}f(x,y)=lim_{y ightarrow0}lim_{x ightarrow0}f(x,y)=0$

$ext{But}lim_{(x,y) ightarrow(0,0)}f(x,y) ext{does not exist.}$

5. a) Show that the following function f is differentiable at (0,0):

$f(x,y)=egin{cases}frac{x^2 y^2}{x^2+y^2},& ext{if}(x,y) e(0,0)\0,& ext{if}(x,y)=(0,0)end{cases}$

b) Find the complete solution of the equation px + qy = pq.

6. a) Using Charpit’s method, find the complete integral of the differential equation:

p²x + q²y = z

b) Find partial derivatives f_x and f_y for the function:

f(x,y) = 5x⁴y² + 6x²y³

at the point ,(1,-1)

c) State Euler’s theorem for homogeneous functions and verify it for the function:

$u=frac{x^3+y^3}{x+y}$

7. a) The population of a town grows at a rate proportional to the population at any time. Its initial population of 500 increases by 15% in 10 years. What will be the population in 30 years?

b) Show that the following function is not continuous at (0,0):

$f(x,y)=egin{cases}ysin{frac{1}{x}}+xsin{frac{1}{y}},&x e 0,y e 0\1,& ext{otherwise}end{cases}$

BMTC 132 2026 - English

ASSIGNMENT

Course Code: BMTC-132

Assignment Code: BMTC-132/TMA/2026

Maximum Marks: 100

1. State whether the following statements are true or false. Justify your answer with the help of a short proof or a counter example: $(5 \times 2 = 10)$

a) y² is an integrating factor of the differential equation:

$6xy \, dx + (4y + 9x^2) \, dy = 0$ .
b) The solution of the differential equation $\frac{dy}{d} = y$ with $y(0) = 0$ exists, but is not unique.

c) $\sin x \frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0$ in $]0, \pi[$ is a linear homogeneous equation.

d) The solution of the differential equation

$\frac{dy}{dx} = y$ with $y(0) = 0$

exists, but is not unique.

e) The Pfaffian equation $(2xy^2 + 2xy + 2xz^2 + 1)dx + dy + 2z \, dz = 0$ is integrable.

2. a) Apply the method of variation of parameter to solve the differential equation:

$$y'' + 6y' + 9y = \frac{1}{x^3} e^{-3x}, \quad x > 0.$

b) Suppose that a thermometer having a reading of $75^\circ\text{F}$ inside a house is placed outside where the air temperature is $15^\circ\text{F}$ . Two minutes later it is found that the thermometer reading is $30^\circ\text{F}$ . Find the temperature reading T(t) of the thermometer at any time t.

3. a) Find the integral surface of the p.d.e.:

$$(x - y)p + (y - x - z)q = z$

through the circle $z = 1, x^2 + y^2 = 1$ .

b) Solve: $(x^2y - 2xy^2)dx - (x^3 - 3x^2y)dy = 0$ .

4. a) Using Charpit's method, find the complete integral of the p.d.e.:

$$2xz - px^2 - 2qxy + pq = 0.$

b) Using the method of undetermined coefficients, solve the differential equation:

$$(D^3 + 2D^2 - D - 2)y = e^x + x^2.$

c) Solve the differential equation: $\frac{dy}{dx} = (x + y)^2$ .

5. a) A particle falls from rest in a medium in which the resistance is $\lambda v^2$ per unit mass, v being the velocity of the particle at time t. Prove that the distance fallen in time t is $\frac{1}{\lambda} \ln \cosh (t\sqrt{g\lambda})$ , where g is the acceleration due to gravity.

b) Solve: $(y^2 + yz)dx + (z^2 + zx)dy + (x^2 + xy)dz = 0$ .

6. a) Solve: $(D^2 + 5DD' + 5D'^2)z = x \sin (3x - 2y)$ .

b) Solve: $\frac{dx}{y^2 + yz + z^2} = \frac{dy}{z^2 + zx + x^2} = \frac{dz}{x^2 + xy + y^2}$ .

7. a) Using the method of variation of parameters, solve the equation

$$\frac{d^2y}{dx^2} + a^2y = \sec ax.$

b) Solve $(p + q)(px + qy) = 1$ , using Charpit's method.

c) Solve: $\frac{dy}{dx} = \frac{1}{x + y + 1}$ .

8. a) Solve: $(D^2 - DD' - 2D) z = \sin(3x + 4y) + e^{2x+y}$ .

b) Use the method of variation of parameters to solve the following differential equation:

$$y'' - 2y' + y = \frac{12e^x}{x^3}.$

9. a) Solve: $x^2y'' - 2xy' - 4y = x^2 + 2\ln x$ .

b) Solve the equation $(7y - 3x + 3)dy + (3y - 7x + 7)dx = 0$ .

10. a) Using the method of undetermined coefficients, solve the equation

$$\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = 4x^2.$

b) Using Charpit's method, solve the equation

$$zp^2 - y^2p + y^2q = 0.$

BMTC 132 2025 - Hindi

सत्रीय कार्य

पाठ्यक्रम कोड: BMTC-132

सत्रीय कार्य कोड: BMTC-132/TMA/2025

अधिकतम अंक: 100

1. a) अवकल समीकरण :

$frac{dy}{dx}=-frac{2x+y+1}{4x+2y-1}$

का हल प्राप्त कीजिए।

b) युगपत समीकरणों :

$frac{dx}{y-z}=frac{dy}{z-x}=frac{dz}{x-y}$

का हल प्राप्त कीजिए।

2. क) प्राचल विचार विधि से अवकल समीकरण :

y" + y = sec² x

का व्यापक हल प्राप्त कीजिए।

b) यदि :

Z=x²y+2xy⁴

जहाँ x = sin zt

और y = cost

तो dz/dt ज्ञात कीजिए: (i) श्रृंखला नियम और (ii) प्रत्यक्ष प्रतिस्थापन से जहां t = 0 |

3. a) सत्यापित कीजिए कि अवकल समीकरण :

z(x²-y²-z^²)dx+(x+z)xzdy +x(z²-x²-xy)dz = 0

समाकलनीय है और इसका समाकल भी ज्ञात कीजिए।

b) हल कीजिए :

y'=zy^$+x 4 / xy³

4. a) समीकरण :

x(y²-z²)ux + y(z² -x²)u, +z(x²-y²)u_₂ = 0

का व्यापक हल प्राप्त कीजिए।

b) $f(x,y)=frac{xy}{x^2+y^2}$ द्वारा दिए गए फलन के लिए दिखाइए कि :

Image ignouassignments-ignouacademy-com-ignou-bscfmt-bmtc-132-solved-assignment-html-p-ignou-64808

लेकिन Image ignouassignments-ignouacademy-com-ignou-bscfmt-bmtc-132-solved-assignment-html-p-bmtc-17888 का अस्तित्व नहीं होता।

5. a) दिखाइए कि निम्नलिखित फलन f, (0,0) पर अवकलनीय है :

$f(x,y)=egin{cases}frac{x^2 y^2}{x^2+y^2},& ext{if}(x,y) e(0,0)\0,& ext{if}(x,y)=(0,0)end{cases}$

b) समीकरण px+qy = pq का पूर्ण हल प्राप्त कीजिए।

6. a) चारपिट विधि का उपयोग करते हुए, अवकल समीकरण p²x+q²y = z का पूर्ण समाकल ज्ञात कीजिए।

b) बिन्दु (1,-1) पर फलन f(x,y) = 5x⁴+y² +6x²y³ के लिए, आंशिक अवकलज f_x और f_y ज्ञात कीजिए।

c) समघात फलनों के लिए, ऑयलर प्रमेय का कथन लिखिए तथा फलन $u=frac{x^3+y^3}{x+y}$ के लिए उसका सत्यापन कीजिए।

a) एक शहर की जनसंख्या में किसी (उस) समय की जनसंख्या के समानुपात की दर पर वृद्धि होती है। उसकी प्रारम्भिक जनसंख्या 500 में 10 वर्षों में 15% वृद्धि हो जाती है। 30 वर्षों में यह जनसंख्या कितनी हो जाएगी?

b) दर्शनीय कि फलन :

$f(x,y)=egin{cases}ysinfrac{1}{x}+xsinfrac{1}{y},&x e 0,y e 0\1,end{cases}$ अन्यथा

बिन्दु (0,0) पर संतत नहीं है।

BMTC 132 2026 - Hindi

सत्रीय कार्य

पाठ्यक्रम कोड: BMTC-132
सत्रीय कार्य कोड: BMTC-132/TMA/2026
अधिकतम अंक: 100

1. बताइए कि निम्नलिखित कथन सत्य हैं या असत्य। संक्षिप्त उपपत्ति अथवा प्रत्युदाहरण की सहायता से अपने उत्तर की पुष्टि कीजिए :

क) y² अवकल समीकरण :
$$6xy \, dx + (4y + 9x^2)dy = 0$ $
का समाकलन गुणक है।

ख) अवकल समीकरण $\frac{dy}{d} = y$ , जहाँ $y(0) = 0$ , के हल का अस्तित्व है लेकिन हल अद्वितीय नहीं है।

ग) अंतराल $]0, \pi[$ में समीकरण : $\sin x \frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0$ समघात रैखिक समीकरण है।

घ) अवकल समीकरण
$$\frac{dy}{dx} = y, \quad y(0) = 0$ $
के हल का अस्तित्व है, परंतु हल अद्वितीय नहीं है।

ङ) फैफियन अवकल समीकरण $(2xy^2 + 2xy + 2xz^2 + 1)dx + dy + 2z \, dz = 0$ समाकलनीय है।

2. क) अवकल समीकरण :

$$y'' + 6y' + 9y = \frac{1}{x^3}e^{-3x}, \text{ x} > 0.$ $
को प्राचल विचरण विधि से हल कीजिए।

ख) मान लीजिए एक थर्मामीटर जिसके घर के अंदर रीडिंग $75^{\circ}\text{F}$ है, उसे बाहर रखा जाता है, जहाँ वायु तापमान $15^{\circ}\text{F}$ है। दो मिनट के बाद थर्मामीटर की रीडिंग $30^{\circ}\text{F}$ पाई जाती है। किसी भी समय t पर थर्मामीटर के तापमान की रीडिंग T(t) ज्ञात कीजिए।

3. क) आंशिक अवकल समीकरण :
$$(x - y)p + (y - x - z)q = z$ $
का समाकल पृष्ठ ज्ञात कीजिए जो वृत्त $z = 1, x^2 + y^2 = 1$ से गुजरता हो।

ख) हल कीजिए : $(x^2y - 2xy^2)dx - (x^3 - 3x^2y)dy = 0$ . $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(5)$

4. क) चार्पिट विधि से आंशिक अवकल समीकरण $2xz - px^2 - 2qxy + pq = 0$ का पूर्ण समाकल ज्ञात कीजिए।

ख) अनिर्धारित गुणांक विधि से अवकल समीकरण :
$$(D^3 + 2D^2 - D - 2)y = e^x + x^2.$ $
को हल कीजिए। (4)

ग) अवकल समीकरण $\frac{dy}{dx} = (x + y)^2$ को हल कीजिए।

5. क) एक कण विरामावस्था से एक माध्यम, जिसमें प्रतिरोध $\lambda v^2$ प्रति इकाई द्रव्यमान है, में नीचे गिरता है। v किसी भी समय t पर कण का वेग है। सिद्ध कीजिए कि कण द्वारा समय t में तय की गई दूरी $\frac{1}{\lambda} \ln \cosh (t \sqrt{g\lambda})$ है, जहाँ g गुरुत्वीय त्वरण है।

ख) हल कीजिए : $(y^2 + yz)dx + (z^2 + zx)dy + (y^2 - xy)dz = 0.$

6. क) हल कीजिए : $(D^2 + 5DD' + 5D'^2)z = x \sin (3x - 2y).$

ख) हल कीजिए : $\frac{dx}{y^2 + yz + z^2} = \frac{dy}{z^2 + zx + x^2} = \frac{dz}{x^2 + xy + y^2}.$ $\quad\quad\quad\quad\quad\quad\quad\quad$

7. क) प्राचल विचरण विधि द्वारा समीकरण
$$\frac{d^2y}{dx^2} + a^2 y = \sec ax$ $
का हल प्राप्त कीजिए। $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

ख) चार्पिट विधि द्वारा समीकरण $(p + q)(px + qy) = 1$ का हल प्राप्त कीजिए। $\quad\quad\quad\quad\quad$
ग) हल कीजिए : $\frac{dy}{dx} = \frac{1}{x + y + 1}.$ $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$ (2)

8. क) हल कीजिए : $(D^2 - DD' - 2D) z = \sin(3x + 4y) + e^{2x+y}.$ $\quad\quad\quad\quad\quad\quad\quad\quad$

ख) प्राचल विचरण विधि से निम्नलिखित अवकल समीकरण: $y'' - 2y' + y = \frac{12e^x}{x^3}$ को हल कीजिए। $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

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