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IGNOU MTE 8 Solved Assignment 2025

This is latest Solved Assignment of MTE 8 of BSC .

Latest 2025 Solved Assignment
Fully Solved MTE 8 2025 Assignment
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MTE 8 ( Differential Equations )
Differential Equations 2025 Solved Assignment
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Last Date of Submission of IGNOU MTE-08 (BSC) 2025 Assignment is for January 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Semester Wise
January 2025 Session: 30th March, 2025 (for June 2025 Term End Exam).
July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).

Title Name	IGNOU MTE 8 Solved Assignment 2025
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	2025
Session
Language	English Medium
Assignment Code	MTE-08/Assignmentt-1//2025
Product Description	Assignment of BSC (Bachelor in Science) 2025. Latest MTE 08 2025 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MTE-08 (BSC) 2025 Assignment is for January 2025 Session: 30th September, 2025 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2025 (for June 2025 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).

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Questions Included in this Help Book

Ques 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 $\frac{\partial^2 u}{\partial y\partial x}=x^3-y$ के हल में दो स्वेच्छ अचर शामिल होंगे।

Ques 2.

प्राचल विचार विधि से गुणांक

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

को हल कीजिए।

Ques 3.

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।

Ques 4.

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।

Ques 5.

समीकरण y" + w²y = 0 के संगत रिकेटी समीकरण ज्ञात कीजिए।

Ques 6.

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

Ques 7.

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

Ques 8.

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

Ques 9.

यदि ƒ और g स्वेच्छ फलन हों तो दिखाइए कि $u=f(x-vt+i\alpha y)+g(x-vt+i\alpha y),$ 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, $\alpha^2=1-\frac{v^2}{c^2}.$

Ques 10.

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

Ques 11.

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।

Ques 12.

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

Ques 13.

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

Ques 14.

निम्नलिखित सीमा मान समस्या को हल कीजिए

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

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

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

Ques 15.

आंशिक अवकल समीकरण

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

को हल कीजिए।

Ques 16.

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

Ques 17.

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

Ques 18.

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

Ques 19.

सत्यापित कीजिए कि समीकरण

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

समाकलनीय है और इसका पूर्वग ज्ञात कीजिए।

Ques 20.

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?

Ques 21.

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।

Ques 22.

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

Ques 23.

निम्नलिखित अवकल समीकरण को हल कीजिए।

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

Ques 24.

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।

Ques 25.

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।

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

Ques 26.

जैकोबी विधि से समीकरण

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

का पूर्ण समाकल ज्ञात कीजिए।

Ques 27.

हल कीजिए :

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

Ques 28.

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 $\frac{\partial^2 u}{\partial y\partial x}=x^3-y$ involves two arbitrary constants.

Ques 29.

Using the method of variation of parameters, solve the equation

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

Ques 30.

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.

Ques 31.

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

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

Ques 32.

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

Ques 33.

Find the general solution of the DE

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

Ques 34.

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

Ques 35.

If ƒ and g are arbitrary functions of their respective arguments, show that $u=f(x-vt+i\alpha y)+g(x-vt+i\alpha y),$ 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, $\alpha^2=1-\frac{v^2}{c^2}.$

Ques 36.

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

Ques 37.

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.

Ques 38.

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

Ques 39.

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

Ques 40.

Solve the following boundary value problem

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

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

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

Ques 41.

Solve the PDE

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

Ques 42.

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

Ques 43.

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

Ques 44.

Solve: $(1+x)^2\frac{d^2y}{dx^2}+(1+x)\frac{dy}{dx}+y=4\cos\ln(1+x)$

Ques 45.

Verify that the equation

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

is integrable and find its primitive.

Ques 46.

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?

Ques 47.

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

Ques 48.

Find the complete integral of p² +q²-2px-2qy+1 = 0

Ques 49.

Solve the differential equation

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

Ques 50.

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

Ques 51.

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.

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

Ques 52.

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

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

Ques 53.

Solve:

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

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Course Name	Bachelor in Science
Course Code	BSC
Programm	BACHELOR DEGREE PROGRAMMES Courses
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IGNOU MTE 8 Solved Assignment

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