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IGNOU BSC MTE 8 Solved Assignment 2024

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Last Date of Submission of IGNOU MTE-08 (BSC) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Semester Wise
January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam).
July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).

Title Name	IGNOU MTE 8 2024 Solution
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	2024
Session	-
Language	English Medium
Assignment Code	MTE-08/Assignmentt-1//2024
Product Description	Assignment of BSC (Bachelor in Science) 2024. Latest MTE 08 2024 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MTE-08 (BSC) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam). Semester Wise January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam). July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Assignment Code	MTE 8/2024

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Rs. 50

Questions Included in this Help Book

Ques 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

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

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 $x^2+2y^2=c^.$

iii) The normal form of the differential equation

$y^{''}-4xy'+(4x^2-1)y=-3ex^2 sin 2x is \frac{d^2v}{dx} +v=-3sin2x,$

where $v=ye^{-x2}.$

iv The solution of the pde $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=z^2$ is $z=[y+f(x-y)].$

Ques 2.

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

Ques 3.

a) Solve $\frac{dy}{dx}+xy=y^2e^{x2/2}$ sin x.

Ques 4.

b) Write the ordinary differential equation $y\,dx+(xy+x-3y)dy=0$

in the linear form, and hence find its solution.

Ques 5.

c) Given that $y_1(x)=x^{-1}$ is one solution of the differential equation $2x^2y''+3xy'-y=0,\,x>0,$

find a second linearly independent solution of the equation.

Ques 6.

a) Solve, using the method of variation of parameters $\frac{d^2y}{dx^3}-y=\frac{2}{1+e^x}.$

Ques 7.

b) Solve the following equation by changing the independent variable $(1+x^2)^2y''+2x(1+x^2)y'+4y=0.$

Ques 8.

a) Find the integrating factor of the differential equation

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

and hence solve it.

Ques 9.

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

Ques 10.

c) Solve the following $IVP$ $\frac{d^2y}{dx^2}+\frac{dy}{dx}-2y=-6sin\,2x-18cos2x$

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

Ques 11.

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

Ques 12.

b) Find the charge on the capacitor in an RLC circuit at $t=0.0$ sec.when $L=0.05$ Henry, $R=2$ ohms, $C=0.01$ Farad. $E(t)=0,q(0)=5$ Columbus and $i(0)=0.$

Ques 13.

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

Ques 14.

a) Solve the following DEs

(i) $\left ( \frac{dy}{dx}-1 \right )^2\left ( \frac{d^2y}{dx2}+1 \right )^2\,y=sin^2(x/2)+e^x+x.$ (ii) $2x^2y\left ( \frac{d^2y}{dx^2} \right )+4y^2=x^2\left ( \frac{dy}{dx} \right )^2+2xy\left ( \frac{dy}{dx} \right ).$

Ques 15.

b) The differential equation of a damped vibrating system under the action of an external periodic force is: $\frac{d^2x}{dt^2}+2m_0\frac{dx}{dt}+n^2x=a$ $\frac{d^2x}{dt^2}+2m_0\frac{dx}{dt}+n^2x=a\,cos\,pt$

Show that, if $n>m_0>0$ the complementary function of the differential equation represents vibrations which are soon damped out. Find the particular integral in terms of periodic functions.

Ques 16.

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

Ques 17.

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

Ques 18.

b) Solve the following equation by Jacobi’s method $x^2\frac{\partial u}{\partial x}-\left ( \frac{\partial u}{\partial y} \right )^2-a\left ( \frac{\partial u}{\partial z} \right )^2=0.$