~~Rs.~~
Rs. 50

ignou MMT 7 solved 2024

~~Rs.~~
Rs. 50

Last Date of Submission of IGNOU MMT-07 (MSCMACS) 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 MMT 7 solved assignment 2024
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCMACS
Course Name	M.Sc. Mathematics with Applications in Computer Science
Subject Code	MMT 7
Subject Name	Differential Equations and Numerical Solutions
Year	2024
Session	-
Language	English Medium
Assignment Code	MMT-07/Assignmentt-1//2024
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2024. Latest MMT 07 2024 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MMT-07 (MSCMACS) 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	MMT 7/2024

~~Rs.~~
Rs. 50

Questions Included in this Help Book

Ques 1.

Show that $f(x,y)=xy$
i) satisfies a Lipschitz condition on any rectangle $a \leq x \leq b$ and $c \leq y \leq d$ ;

ii) satisfies a Lipschitz condition on any strip $a \leq x \leq b$ and $- \infty < y < \infty$ ;

iii) does not satisfy a Lipschitz condition on the entire plane.

Ques 2.

Use Frobenious method to find the series solution about $x=0$ of the equation
$x(1-x)\frac{d^2y}{dx^2}-(1+3x)\frac{dy}{dx}-y = 0$

Ques 3.

For the following differential equation locate and classify its singular points on the x-axis
i) $x^3 (x-1)y'' - 2(x-1)y' + 3xy = 0$
ii) $(3x+1)xy'' - (x+1)y' +2y = 0$

Ques 4.

Show that
$L_{n+1}(x) = (2n+1-x)L_{n}(x) - n^2 L_{n-1}(x)$

Ques 5.

Show that
$\int_{-1}^{1}x^2P_{n-1}(x)P_{n+1}(x)dx = \frac{2n(n+1)}{(2n-1)(2n+1)(2n+3)}$

Ques 6.

Construct Green's function for the differential equation
$xy'' + y' = 0, 0 < x < l$

under the conditions that y(0) is bounded and y(l)=0.

Ques 7.

Show that between every successive pair of zeros of $J_0(x)$ there exists a zero of $J_1(x)$ .

Ques 8.

Using the transformation $y=x^{1/2}u, 2x^{3/2} = 3z$ , find the solution of $y'' + xy = 0$ in terms of Bessel's functions.

Ques 9.

Show that $\int_{0}^{\infty} e^{-ax}J_0(bx)dx = \frac{1}{\sqrt{a^2+b^2}}, a>0, b>0$

Ques 10.

Find the Laplace transform of $\frac{cos\sqrt{t}}{\sqrt{t}}$ .

Ques 11.

If $k_m$ and $k_n$ are distinct roots of Bessel function $J_p(kb) = 0$ with $p\geq0$ , $b>0$ then show that $\int_{0}^{b}x J_p(k_mx)J_p(k_nx)dx = \left\{\begin{matrix} 0 & if m\neq n\\ \frac{b^2}{2}[J_{p+1}(k_nb)] & if \: m = n \end{matrix}\right.$

Ques 12.

Solve the following IBVP using Laplace transform technique:
$u_t = u_{xx} , 0<x<1, t>0$
$u(0,t) = 1, u(1,t) = 1, t>0$
$u(x,0) = 1 + sin \pi x, 0<x<1$

Ques 13.

If the Fourier cosine transform of $f(x)$ is $\alpha^ne^{-a\alpha}$ , then show that

$f(x) = \frac{2}{\pi}\frac{n!cos(n+1)\theta)}{(a^2+x^2)^{\frac{n+1}{2}}}$

Ques 14.

If the Fourier cosine transform of $f(x)$ is $\alpha^ne^{-a\alpha}$ , then show that

$f(x) = \frac{2}{\pi}\frac{n!cos(n+1)\theta)}{(a^2+x^2)^{\frac{n+1}{2}}}$

Ques 15.

Find the displacement $u(x,t)$ of an infinite string using Fourier transform method given that the string is initially at rest and the initial displacement is $f(x)$ , $-\infty < x < \infty$ .

Ques 16.

Using Fourier integral representation show that $\int_{0}^{\infty}\frac{cos(\alpha x) + \alpha sin(\alpha x)}{1 + \alpha^2} = \left\{\begin{matrix} 0 & if \: x<0\\ \pi/2 & if \: x=0 \\ \pi e^{-x} & if \: x>0 \end{matrix}\right.$

Ques 17.

Using Runge-Kutta 2nd order method with

(i) $h=0.1$ , (ii) $h=0.2$ , solve the initial value problem
$\begin{align*} y'&=y^2\sin x , y(0) =1 \end{align*}$ upto $x=0.4$ . If exact solution is $y = sec x$ , obtain the error.

Ques 18.

Solve heat equation $u_{tt}=u_{xx}$ in $R(0 \leq x \leq 1, t > 0)$ with conditions $u(x,0)=0, u(0,t)=0, u(1,t)=t$ using Crank-Nicolson method with $h=0.25$ , $\lambda=1$ upto two time steps.

Ques 19.

Using second order finite difference method, solve the boundary value problem $y'' + 5y' + 4y= 1$ , $y(0) = 0$ , $y(1) = 0$ , $h=1/4$

Ques 20.

Solve wave equation $u_{tt}=u_{xx}$ with
$u(x,0) = 0 , u_t(x,0)=0, u(0,t)=0 , u(1,t) = 100 sin(\pi t)$ with $k=h=0.25$ , using explicit method upto 4 time levels.

Ques 21.

Find approximate value of $y(0,1)$ for initial value problem
$y'=x^3-y^3 , y(0)=1$ using multiple method

$\begin{equation} y_{n+1} = y_n + \frac{h}{3}(7f_n - 2f_{n-1} + f_{n-2}) \end{equation}$

with $h=0.2$ . Calculate the starting values using Runge-Kutta second order method with the same h.

Ques 22.

Using standard five-point formula, solve Laplace equation $\nabla^2u=0$ in R where R is the square $0 \leq x \leq 1, 0 \leq y \leq 1$ subject to the boundary conditions $u(x,y) = x^2 - y^2$ on $x=0 , y=0, y=1$
and $3u + 2\frac{\partial u}{\partial x} = x^2 + y^2$ on $x=1$ . Assume $h=k=1/2$ .

Ques 23.

Find approximate value of $y(1.0)$ for the initial value problem $\begin{align*} y' =x-2y, y(0)=1 \end{align*}$ using Milne-Simpson's method $y_{n+1} = y_{n-1} + \frac{h}{3}[f_{n+1} + 4f_{n} - f_{n-1}]$
with $h=0.2$ . Calculate starting value using Runge-Kutta fourth order method with the same h.