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MTE 8: Differential Equations

Title Name	IGNOU MTE 8 SOLVED ASSIGNMENT HINDI
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 8/Assignment-1/2025
Product Description	Assignment of BSC (Bachelor in Science) 2025. Latest MTE-08 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).
Format	Ready-to-Print PDF (.soft copy)

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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 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 - 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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