IGNOU BSC MTE 8 Differential Equations Solved Assignment English

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MTE 8: Differential Equations

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Title Name	IGNOU BSC MTE 8 Differential Equations Solved Assignment English
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 8
Subject Name	Differential Equations
Year	2026
Session	-
Language	English
Assignment Code	MTE 8/Assignment-1/2026
Product Description	Assignment of BSC (Bachelor in Science) 2026. Latest MTE 08 2026 Solved Assignment Solutions
Format	Ready-to-Print PDF (.soft copy)

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MTE 8 2026 - English

ASSIGNMENT

Course Code: MTE-08

Assignment Code: MTE-08/TMA/2026

Maximum Marks: 100

1. State whether the following statement are true or false. Justify your answer with the help of a short proof or a counter-example.

i) The initial value problem

$$\frac{dy}{dx} = x^2 + y^2, \quad y(0) = 0$

has a unique solution in some interval of the form -h < x < h.

ii) The orthogonal trajectories of all the parabolas with vertices at the origin and foci on the x-axis is $x^2 + 2y^2 = c^2$ .

iii) The normal form of the differential equation

$$y'' - 4xy' + (4x^2 - 1)y = -3e^{x^2} \sin 2x \text{ is}$

$$\frac{d^2v}{dx^2} + v = -3 \sin 2x, \text{ where } v = ye^{-x^2}.$

iv) The solution of the pde $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = z^2$ is $z = -[y + f(x - y)]^{-1}$ . (Note: The image appears to have a small typo in the solution provided, it should likely be to the power of -1 based on standard solutions, however, transcribing exactly as seen: $z = -[y + f(x - y)]$ .)

v) The pde $u_{xx} + x^2 u_{xy} - \left( \frac{x^2}{2} + \frac{1}{4} \right) u_{yy} = 0$ is hyperbolic in the entire xy-plane.

2. a) Solve $\frac{dy}{dx} + xy = y^2 e^{x^2/2} \sin x$ .

b) Write the ordinary differential equation

$$y \, dx + (xy + x - 3y) \, dy = 0$

in the linear form, and hence find its solution.

c) Given that $y_1(x) = x^{-1}$ is one solution of the differential equation

$$2x^2y'' + 3xy' - y = 0, \quad x > 0,$

find a second linearly independent solution of the equation.

3. a) Solve, using the method of variation of parameters

$$\frac{d^2y}{dx^2} - y = \frac{2}{1 + e^x}.$

b) Solve the following equation by changing the independent variable

$$(1 + x^2)^2 y'' + 2x(1 + x^2) y' + 4y = 0.$

4. a) Find the integrating factor of the differential equation

$$(6xy - 3y^2 + 2y) \, dx + 2(x - y) \, dy = 0$

and hence solve it.

b) Solve the equation $x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = x^m$ , for all positive integer values of m.

c) Solve the following IVP

$$\frac{d^2y}{dx^2} + \frac{dy}{dx} - 2y = -6 \sin 2x - 18 \cos 2x$

$$y(0) = 2, \quad y'(0) = 2.$

5. a) Solve: $\frac{d^2y}{dx^2} - 2 \tan x \frac{dy}{dx} + 5y = e^x \sec x$ .

b) Find the charge on the capacitor in an RLC circuit at $t = 0.01$ sec. when $L = 0.05$ Henry, $R = 2$ ohms, $C = 0.01$ Farad. $E(t) = 0, q(0) = 5 \text{ Columbus and } i(0) = 0$ .

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

6. a) Solve the following DEs
(i) $\left( \frac{dy}{dx} - 1 \right)^2 \left( \frac{d^2y}{dx^2} + 1 \right)^2 y = \sin^2(x/2) + e^x + x$ .

(ii) $2x^2y \left( \frac{d^2y}{dx^2} \right) + 4y^2 = x^2 \left( \frac{dy}{dx} \right)^2 + 2xy \left( \frac{dy}{dx} \right)$ .

b) The differential equation of a damped vibrating system under the action of an external periodic force is:

$$\frac{d^2x}{dt^2} + 2m_0 \frac{dx}{dt} + n^2x = a \cos pt$

Show that, if n > m₀ > 0 the complementary function of the differential equation represents vibrations which are soon damped out. Find the particular integral in terms of periodic functions.

7. a) Verify that the Pfaffian differential equation

$$yz \, dx + (x^2y - zx) \, dy + (x^2z - xy) \, dz = 0$

is integrable and hence find its integral.

b) Solve the following equation by Jacobi's method

$$x^2 \frac{\partial u}{\partial x} - \left( \frac{\partial u}{\partial y} \right)^2 - a \left( \frac{\partial u}{\partial z} \right)^2 = 0.$

c) Show that $2z = (ax + y)^2 + b$ , where a, b are arbitrary constants is a complete integral of $px + qy - q^2 = 0$ .

8. a) Solve the following differential equations

(i) $x^2p + y^2q = (x + y) z$ .

(ii) $\sqrt{p} - \sqrt{q} + 3x = 0$ .

b) Find the equation of the integral surface of the differential equation

$$(x^2 - yz) \, p + (y^2 - zx) \, q = z^2 - xy$

which passes through the line $x = 1, y = 0$ .

9. a) Using the method of separation of variables, solve $u_{xt} = e^{-t} \cos x$ when

$$u(x, 0) = 0, \frac{\partial u}{\partial t}(0, t) = 0.$

b) Find the temperature in a bar of length $\ell$ with both ends insulated and with initial temperature in the rod being $\sin \frac{\pi x}{\ell}$ .

10. a) Solve the following differential equations

(i) $[D^3 - DD'^2 - D^2 + DD'] \, z = 0$ .

(ii) $[D^4 - D'^4 - 2D^2 D'^2] \, z = 0$ .

(iii) $[D^2 - 2DD' + D'^2] \, z = 12xy$ .

b) Show that the wave equation $a^2 u_{xx} = u_{tt}$ can be reduced to the form $u_{\xi\eta} = 0$ by the change of variable $\xi = x - at, \eta = x + at$ .

MTE 8 2025 - English

Assignment (MTE-08)
Course Code: MTE-08
Assignment Code: MTE-08/TMA/20255
Maximum Marks: 100

1. Classify the following statements as true or false. Give a short proof of a counter example in support of your answer.

i) The solution of the differential equation $frac{dy}{dx}=y,quad y(0)=0$ exists, but is not unique.

ii) The differential equation representing all tangents ty = x + t² at the point (t², 2t) to the parabola y² = 4x is x(y')² + yy'+1=0

iii) The p.d.e. au_xx + 2b u_xy + cu,_yy = 0 where a, are constants is irreducible when b² ac = 0

iv) The functions f₁(x) = cos²x, f₂(x) = sin²x, f₃(x) = sec² x and f₄(x) = tan²x are linearly dependent on the interval ] - π/2, π/2 [

v) The solution of the second order partial differential equation $frac{partial^2 u}{partial ypartial x}=x^3-y$ involves two arbitrary constants.

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

$frac{d^2y}{dx^2}+y=operatorname{cosec}x,quad 0le x<frac{pi}{2}$

b) A mass m , free to move along a line is attracted towards a given point on the line with a force proportional to its distance from the given point. If the mass starts from rest at a distance x₀ from the given point, show that the mass moves in a simple harmonic motion.

c) Show that, for the differential equation

$frac{d^2y}{dx^2}+a(x)frac{dy}{dx}+b(x)y=0$

e^mx is a particular integral if m² + am+b = 0. Hence find the value of m so that e^xm is a particular integral of the equation

$(x-2)frac{d^2y}{dx^2}-(4x-7)frac{dy}{dx}+(4x-6)y=0$

3. a) Obtain the Riccati equation associated with the equation y" + w²y = 0 and hence find its solution.

b) Solve the differential equation $frac{d^2y}{dx^2}=a+bx+cx^2$ , given that $frac{dy}{dx}=0.$ and y =d

c) Find the general solution of the DE

y² ln y = x py + p² Does the equation has any singular solution? If yes, obtain it.

घ) Reduce the equation x² (y - px) = yp² to clairaut’s form and hence find its complete solution.

4. क) If ƒ and g are arbitrary functions of their respective arguments, show that $u=f(x-vt+ialpha y)+g(x-vt+ialpha y),$ is a solution of $frac{partial^2 u}{partial x^2}+frac{partial^2 u}{partial y^2}=frac{1}{c^2}frac{partial^2 u}{partial t^2}$ where, $alpha^2=1-frac{v^2}{c^2}.$

ख) Solve the following differential equations:

i) $frac{dx}{y^2(x-y)}=frac{dy}{x^2(x-y)}=frac{dz}{z(x^2+y^2)}$

ii) $(yz+z^2)dx-xzdy+xydz=0$

iii) $(z+z^3)cos x,dx-(z+z^3)dy+(1-z^2)(y-sin x)dz=0.$

5. क) if y₁ = 2x + 2 and y₂ = -x²/2 are the solutions of the equation y = xy' + (y')²/2 then are the constant multiples c₁y₁, and c₂y₂, where c₁ and c₂ are arbitrary, also the solutions of the given DE? Is the sum y₁ + y₂ a solution? Justify your answer.

b) Find the orthogonal trajectories of the family of parabolas x = cy² and sketch their graph.

c) Find the general solution of the differential equation $frac{y^2}{2}+2ye^t+(y+e^t)frac{dy}{dt}=0$

6. a) Solve the following boundary value problem

$u_t=u_{xx},quad 0<x<l,quad t>0$

$u(0,t)=u(l,t)=0$

$u(x,0)=x(l-x),quad 0le xle l$

b) Solve the PDE

$x^2frac{partial^2 z}{partial x^2}-y^2frac{partial^2 z}{partial y^2}+xfrac{partial z}{partial x}-yfrac{partial z}{partial y}=ln x$

7. a) Solve : $frac{d^2y}{dx^2}-4xfrac{dy}{dx}+(4x^2-1)y=-3e^{x^2}sin 2x.$

b) Solve the IVP : $x^2frac{dy}{dx}=cos x-2xy,quad y(pi)=0,quad x>0.$

ग) Solve: $(1+x)^2frac{d^2y}{dx^2}+(1+x)frac{dy}{dx}+y=4cosln(1+x)$

8. a) Verify that the equation

$2(y+z)dx-(x+z)dy+(2y-x+z)dz=0$

is integrable and find its primitive.

b) Find the differential equation of the family of surfaces $phi(x+y+z,x^2+y^2-z^2)=0$ What is the order of this p.d.e?

c) Find the integral surface of the linear p.d.e. x(y² + z)p-y(x² + z)q = (x²- y²) z which contains the straight line x + y = 0, z = 1

9. a) Find the complete integral of p² +q²-2px-2qy+1 = 0

b) Solve the differential equation

$p^2+2pycot x-y^2=0$ where $p=frac{dy}{dx}.$

c) Find the directional derivatives of f(x, y, z) = x² + 2y² +3z² at P_o (1, 1, 1) in the direction of a =i+j+k

10. a) Find the deflection of the fixed end vibrating string of unit length corresponding to zero initial deflection and ) u(x given below as the initial velocity.

$u(x)=egin{cases}x,&0le x<frac{1}{2}\1-x,&frac{1}{2}le x<1end{cases}$

b) Using Jacobi’s method find the complete integral of the equation

$2u_1 xz+3u_2 y^2+u_2^2 u_3=0$

c) Solve:

$(D^2-2DD'+D'^2)z=12xy$

MTE 8 2025 - Hindi

सत्रीय कार्य

पाठ्यक्रम कोड: MTE-08

सत्रीय कार्य कोड: MTE08/TMA/2025

अधिकतम अंक 100

1. बताइए कि निम्नलिखित कथन सत्य है या असत्य? अपने उत्तर की पुष्टि उपपत्ति या प्रति उदाहरण की सहायता से कीजिए।

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

ii) परवलय y² = 4x के बिंदु (t², 2t) पर सभी स्पर्श रेखाओं ty = x + t² को निरूपित करने वाला अवकल समीकरण x(y')² + yy'+1=0 है।

iii) आंशिक अवकल समीकरण au_xx + 2b u_xy + cu,_yy = 0 जहाँ a, b, c अचर हैं, असमनिय होता है जब b² ac = 0 हो।

iv) अंतराल ] - π/2, π/2 [ में फलन f₁(x) = cos²x, f₂(x) = sin²x, f₃(x) = sec² x तथा f₄(x) = tan²x रैखिकतः परतंत्र है।

v) द्वितीय कोटि आंशिक अवकल समीकरण $frac{partial^2 u}{partial ypartial x}=x^3-y$ के हल में दो स्वेच्छ अचर शामिल होंगे।

2. क) प्राचल विचार विधि से गुणांक

$frac{d^2y}{dx^2}+y=operatorname{cosec}x,quad 0le x<frac{pi}{2}$

को हल कीजिए।

ख) एक द्रव्यमान m जो मुक्त रूप से एक रेखा पर गतिशील है वह रेखा पर दिए गए एक बिन्दु की ओर उस बिन्दु से अपनी दूरी के समानुपाती बल से आकृष्ट होता है। यदि द्रव्यमान दिए गए बिन्दु से दूरी x₀ पर विश्रामावस्था से प्रारंभ होता है तो दिखाइए कि द्रव्यमान सरल आवर्त गति में गतिमान होता है।

ग) दिखाइए कि अवकल समीकरण

$frac{d^2y}{dx^2}+a(x)frac{dy}{dx}+b(x)y=0$

के लिए e^mx विशेष समाकल है यदि m² + am+b = 0. अतः m का वह मान ज्ञात कीजिए जिसके लिए e^xm समीकरण

$(x-2)frac{d^2y}{dx^2}-(4x-7)frac{dy}{dx}+(4x-6)y=0$

का एक विशेष समाकल हो।

3. क) समीकरण y" + w²y = 0 के संगत रिकेटी समीकरण ज्ञात कीजिए।

ख) अवकल समीकरण $frac{d^2y}{dx^2}=a+bx+cx^2$ को हल कीजिए जबकि दिया गया है कि और $frac{dy}{dx}=0.$

ग) अवकल समीकरण y² ln y = x py + p² का व्यापक हल ज्ञात कीजिए। क्या समीकरण का कोई विचित्र हल है?

घ) समीकरण x² (y - px) = yp² को क्लेरों रूप में समानीत कीजिए और फिर इसका पूर्ण हल ज्ञात कीजिए।

4. क) यदि ƒ और g स्वेच्छ फलन हों तो दिखाइए कि $u=f(x-vt+ialpha y)+g(x-vt+ialpha y),$ समीकरण $frac{partial^2 u}{partial x^2}+frac{partial^2 u}{partial y^2}=frac{1}{c^2}frac{partial^2 u}{partial t^2}$ का हल होगा जहाँ, $alpha^2=1-frac{v^2}{c^2}.$

ख) निम्नलिखित अवकल समीकरणों के हल ज्ञात कीजिए

i) $frac{dx}{y^2(x-y)}=frac{dy}{x^2(x-y)}=frac{dz}{z(x^2+y^2)}$

ii) $(yz+z^2)dx-xzdy+xydz=0$

iii) $(z+z^3)cos x,dx-(z+z^3)dy+(1-z^2)(y-sin x)dz=0.$

5. क) यदि y₁ = 2x + 2 और y₂ = -x²/2 समीकरण y = xy' + (y')²/2 के दो हल हों तो क्या इनके अचर गुणज c₁y₁, और c₂y₂, जहाँ c₁ और c₂ स्वेच्छ अचर हैं, भी समीकरण के हल होंगें? क्या इनका योगफल y₁ + y₂ एक हल है? अपने उत्तर की पुष्टि कीजिए।

ख) परवलय कुल x = cy² की लंबकोणीय संद्देदियाँ ज्ञात कीजिए और उनके ग्राफ बनाइए।

ग) अवकल समीकरण $frac{y^2}{2}+2ye^t+(y+e^t)frac{dy}{dt}=0$ का व्यापक हल प्राप्त कीजिए।

6. क) निम्नलिखित सीमा मान समस्या को हल कीजिए

$u_t=u_{xx},quad 0<x<l,quad t>0$

$u(0,t)=u(l,t)=0$

$u(x,0)=x(l-x),quad 0le xle l$

ख) आंशिक अवकल समीकरण

$x^2frac{partial^2 z}{partial x^2}-y^2frac{partial^2 z}{partial y^2}+xfrac{partial z}{partial x}-yfrac{partial z}{partial y}=ln x$

को हल कीजिए।

7. क) हल कीजिए: $frac{d^2y}{dx^2}-4xfrac{dy}{dx}+(4x^2-1)y=-3e^{x^2}sin 2x.$

ख) हल कीजिए : $x^2frac{dy}{dx}=cos x-2xy,quad y(pi)=0,quad x>0.$

ग) प्राचल विचरण विधि से समीकरण $(1+x)^2frac{d^2y}{dx^2}+(1+x)frac{dy}{dx}+y=4cosln(1+x)$ हल कीजिए। को

8. क) सत्यापित कीजिए कि समीकरण

$2(y+z)dx-(x+z)dy+(2y-x+z)dz=0$

समाकलनीय है और इसका पूर्वग ज्ञात कीजिए।

ख) पृष्ठ-कुल $phi(x+y+z,x^2+y^2-z^2)=0$ का अवकल समीकरण ज्ञात कीजिए। इस आंशिक अवकल समीकरण की कोटि क्या है?

ग) रैखिक आंशिक अवकल समीकरण x(y² + z)p-y(x² + z)q = (x²- y²) z का वह समाकल पृष्ठ ज्ञात कीजिए जिसमें सरल रेखा x + y = 0, z = 1 आविष्ट हो।

9. क) समीकरण p² +q²-2px-2qy+1 = 0 का पूर्ण समाकल ज्ञात कीजिए।

ख) निम्नलिखित अवकल समीकरण को हल कीजिए।

$p^2+2pycot x-y^2=0$ जहां $p=frac{dy}{dx}.$

ग) सदिश a =i+j+k की दिशा में P_o (1, 1, 1) पर f(x, y, z) = x² + 2y² +3z² के दिक्-अवकलज ज्ञात कीजिए।

10. क) शून्य प्रारंभिक विक्षेपण के संगत एकक लंबाई वाली कंपायमान डोरी के नियत सिरे का विक्षेपण ज्ञात कीजिए जहां, नीचे दिया गया u(x) प्रारंभिक वेग है।

$u(x)=egin{cases}x,&0le x<frac{1}{2}\1-x,&frac{1}{2}le x<1end{cases}$

ख) जैकोबी विधि से समीकरण

$2u_1 xz+3u_2 y^2+u_2^2 u_3=0$

का पूर्ण समाकल ज्ञात कीजिए।

ग) हल कीजिए :

$(D^2-2DD'+D'^2)z=12xy$

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Instead of repeatedly searching for answers across different websites and sources, keep your MTE 8 Differential Equations Solved Assignment 2026 ready for reference whenever you study.

A Useful Reference for Your IGNOU Assignment Preparation

Understand the question → Find the relevant points → Study the answer → Prepare your own assignment with confidence.

IGNOU BSC Assignments 2026

The BSC (Bachelor in Science) programme includes assignment work as part of the academic requirements applicable to students. Students should always verify the latest assignment questions, instructions and submission requirements from the official IGNOU website.

Our MTE 8 Solved Assignment is independently prepared reference material intended to assist students with understanding and preparing their answers. It is not an official IGNOU publication.

Why Keep a Solved Assignment Reference?

A useful reference is more than simply a collection of answers. It can help you see how a question can be approached, what important points need attention and how the subject matter can be organised into a clear answer.

Save preparation time with organised reference material
Understand difficult questions more easily
Review important concepts before writing your answers
Improve answer organisation with a structured reference
Study conveniently whenever and wherever you need it

Product Information

Course Name	Bachelor in Science
Course Code	BSC
Subject Code	MTE 8
Subject	Differential Equations
Programme	BACHELOR DEGREE PROGRAMMES Courses
Language	English
Material Type	Solved Assignment – Soft Copy / PDF

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Get the 2026 MTE 8 Differential Equations Solved Assignment as a convenient reference for your assignment preparation.

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IGNOU MTE 8 2026 Solved Assignment
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Important Note for IGNOU Students

Always check the latest assignment questions and submission instructions on the official IGNOU website before preparing and submitting your assignment. Use this solved assignment as a study and reference resource and prepare your own submission according to the applicable IGNOU requirements.

Why Choose IGNOU Academy for Your Assignments?

Getting your assignments right is an important part of your IGNOU academic journey.IGNOU Academy provides carefully prepared reference materials designed to makeassignment preparation simpler, clearer and more convenient for students.

Latest Curriculum: Materials are prepared with the current IGNOU syllabus and assignment requirements in mind.
Proper Assignment Format: Understand the structure, presentation and answer style generally expected in an IGNOU assignment.
Simple Explanations: Difficult topics are explained in clear and easy-to-understand language.
Exam Preparation: Assignment material can also be useful for revising important concepts before the Term-End Examination.
Student-Friendly Language: Clear English/Hindi content helps students understand topics without unnecessary complexity.
Trusted by IGNOU Learners: Students across India use IGNOU Academy as a convenient source of assignment and study-reference material.

These materials are provided as reference study guides. Students should understandthe concepts, follow IGNOU guidelines and prepare their own original assignmentswhere required.

How to Get Your Solved Assignment PDF

Visit IGNOU Academy:Go to www.ignouacademy.com.
Find Your Course:Search for your programme, course code and subject.
Select the Session:Choose the latest available reference material for your academic session.
Complete Your Order:Add the required material to your cart, log in or register, and complete the payment.
Access Your Material:After successful payment, access your purchased study material from your account.

Step-by-Step: Finding IGNOU Assignment Question Papers

Open the Website:Visit www.ignouacademy.com.
Open Assignment Question Papers:Select the relevant IGNOU assignment question paper section.
Filter Your Requirement:Select your course, session and medium such as English or Hindi.
Open the Paper:View or download the required question paper for your preparation.

How to Submit Your IGNOU Assignments

Prepare Neatly:Use the format and submission instructions applicable to your programme and study centre.
Complete the Front Page:Enter your Name, Enrollment Number, Programme/Course Code, Subject and Study Centre details correctly.
Offline Submission:If your study centre requires physical submission, submit the assignment as instructed and retain the acknowledgement.
Online Submission:Where online submission is permitted, follow the instructions issued by your Regional Centre or Study Centre. Keep proof of submission for your records.

Tracking Your Assignment Submission Status

After submitting your assignments, students can check whether their assignmentmarks and submission details have been updated through the applicable official IGNOU system.

Open the Official IGNOU Student Area:Visit the official IGNOU website and access the relevant student services.
Find Assignment Status:Open the assignment-status service applicable to your programme.
Enter Your Details:Provide your Enrollment Number and other requested programme details.
Check the Status:Review the available submission, evaluation and marks information.

A Quick Tip for IGNOU Students

Assignments are an important part of the IGNOU evaluation process. Preparing themcarefully, understanding the questions, following the prescribed format and submittingthem within the applicable deadline can help you stay on track academically.

Smart Study + Proper Preparation + Timely Submission = Better Academic Planning

Need Help? Contact IGNOU Academy WhatsApp: +91 9199852182 | Website: www.ignouacademy.com

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