IGNOU MMT 7 SOLVED ASSIGNMENT

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IGNOU MMT 7 SOLVED ASSIGNMENT

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IGNOU MMT 7 SOLVED ASSIGNMENT

~~Rs. 200~~
Rs. 123

Last Date of Submission of IGNOU MMT-07 (MSCMACS) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).
Semester Wise
January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam).
July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

Title Name	IGNOU MMT 7 SOLVED ASSIGNMENT
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	2026
Session
Language	English Medium
Assignment Code	MMT-07/Assignmentt-1//2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMT 07 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MMT-07 (MSCMACS) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam). Semester Wise January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

~~Rs. 200~~
Rs. 123

Questions Included in this Help Book

Ques 1.

Check whether $f(x, y) = x^2 y^2$ satisfies Lipschitz condition

i) on any rectangle $a \leq x \leq b$ and $c \leq y \leq d$ ;

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

iii) on the entire plane.

Ques 2.

Find the series solution about $x = 1$ of the equation
$x(1 - x) \frac{d^2 y}{dx^2} - (1 + 3x) \frac{dy}{dx} - y = 0$ .

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

Ques 5.

Construct Green's function for the differential equation

$xy'' + y' = 0, \quad 0 < x < \ell$

under the conditions that y(0) is bounded and $y(\ell) = 0$ .

Based on the image provided, here is the transcription of the mathematical problems:

Ques 6.

how that between every successive pair of zeros of J₀(x) there exists a zero of J₁(x).

Ques 7.

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

Ques 8.

Show that $\int_0^{\infty} e^{-ax} J_0(bx) dx = \frac{1}{\sqrt{a^2 + b^2}}, a > 0, b > 0$ .

Ques 9.

Find the Laplace transform of $\frac{\sin \sqrt{t}}{\sqrt{t}} \dots$

Ques 10.

) If k_m and k_n are distinct roots of Bessel function $J_p(kb) = 0$ with $p \ge 0, b > 0$ then show that
$$\int_0^b x J_p(k_m x) J_p(k_n x) dx = \begin{cases} 0 & \text{if } m \neq n \\ \frac{b^2}{2} [J_{p+1}(k_n b)]^2 & \text{if } m = n \end{cases}$

Ques 11.

Solve the following IBVP using the Laplace transform technique:

$$u_t = 2u_{xx}, \quad 0 < x < 1, \quad t > 0$

$$u(0, t) = 1, \quad u(1, t) = 1, \quad t > 0$

$$u(x, 0) = 1 + \sin \pi x, \quad 0 < x < 1.$

Ques 12.

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

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

Ques 13.

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

Ques 14.

Using Fourier integral representation show that

$$\int_{0}^{\infty} \frac{\cos (\alpha x) + \alpha \sin (\alpha x)}{1 + \alpha^2} d\alpha = \begin{cases} 0 & \text{if } x < 0 \\ \pi / 2 & \text{if } x = 0 \\ \pi e^{-x} & \text{if } x > 0 \end{cases}$

Ques 15.

Using Runge-Kutta second order method with

(i) $h = 0.2$ , (ii) $h = 0.3$ , solve the initial value problem

$$y' = y^2 \sin x, \quad y(0) = 1.$

Upto $x = 0.6$ . If the exact solution is $y = \sec x$ , obtain the error.

Ques 16.

Solve the heat conduction equation $u_t = u_{xx}$ in the region $R(0 \le x \le 1, t > 0)$ with the initial and boundary conditions $u(x, 0) = 0, u(0, t) = 0, u(1, t) = t$ using Crank-Nicolson method with $h = 0.20$ and $\lambda = 1$ upto two time steps.

Based on the third image provided, here is the transcription of the remaining mathematical problems:

Ques 17.

Using second order finite Difference method, solve the boundary value problem

$y'' + 5y' + 4y = 2, y(0) = 0, y(1) = 0, h = 1/2$

Ques 18.

Solve the wave equation $u_{tt} = u_{xx}$ with the initial and boundary conditions

$u(x, 0) = 0, u_t(x, 0) = 0, u(0, t) = 0, u(1, t) = 200 \sin \pi t$ .

with $h = k = 0.25$ , using the explicit method upto four time levels.

Ques 19.

Find an approximate value of y(1.0) for the initial value problem

$y' = x^3 - y^3, y(0) = 1$

using the multiple method

$y_{n+1} = y_n + \frac{h}{3} [7f_n - 2f_{n-1} + f_{n-2}]$

with step length $h = 0.25$ . Calculate the starting values using Runge-Kutta second order method with the same h.

Ques 20.

Using standard five point formula, solve the Laplace equation $\nabla^2 u = 0$ in R where R is the square $0 \le x \le 1, 0 \le y \le 1$ 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 21.

Find an approximate value of y(1.0) for the initial value problem

y = x - 2y, y(0) = 1

using Milne-Simpson’s method

$$y_{n+1} = y_{n-1} + \frac{h}{3} [f_{n+1} + 4f_n + f_{n-1}]$

with the step length $h = 0.2$ . Calculate the starting value using Runge-Kutta fourth order method with the same h.

Ques 22.

) Using fourth order Taylor series method with $h = 0.2$ , solve the initial value problem

$$y' = x + \cos y, \quad y(0) = 0$

upto $x = 1$ .

Ques 23.

Check whether $f(x, y) = x^2 y^2$ satisfies Lipschitz condition

i) on any rectangle $a \leq x \leq b$ and $c \leq y \leq d$ ;

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

iii) on the entire plane.

Ques 24.

Find the series solution about $x = 1$ of the equation
$x(1 - x) \frac{d^2 y}{dx^2} - (1 + 3x) \frac{dy}{dx} - y = 0$ .

Ques 25.

) 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 26.

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

Ques 27.

Construct Green's function for the differential equation

$xy'' + y' = 0, \quad 0 < x < \ell$

under the conditions that y(0) is bounded and $y(\ell) = 0$ .

Based on the image provided, here is the transcription of the mathematical problems:

Ques 28.

how that between every successive pair of zeros of J₀(x) there exists a zero of J₁(x).

Ques 29.

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

Ques 30.

Show that $\int_0^{\infty} e^{-ax} J_0(bx) dx = \frac{1}{\sqrt{a^2 + b^2}}, a > 0, b > 0$ .

Ques 31.

Find the Laplace transform of $\frac{\sin \sqrt{t}}{\sqrt{t}} \dots$

Ques 32.

Ques 33.

Solve the following IBVP using the Laplace transform technique:

$$u_t = 2u_{xx}, \quad 0 < x < 1, \quad t > 0$

$$u(0, t) = 1, \quad u(1, t) = 1, \quad t > 0$

$$u(x, 0) = 1 + \sin \pi x, \quad 0 < x < 1.$

Ques 34.

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

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

Ques 35.

Ques 36.

Using Fourier integral representation show that

Ques 37.

Using Runge-Kutta second order method with

(i) $h = 0.2$ , (ii) $h = 0.3$ , solve the initial value problem

$$y' = y^2 \sin x, \quad y(0) = 1.$

Upto $x = 0.6$ . If the exact solution is $y = \sec x$ , obtain the error.

Ques 38.

Based on the third image provided, here is the transcription of the remaining mathematical problems:

Ques 39.

Using second order finite Difference method, solve the boundary value problem

$y'' + 5y' + 4y = 2, y(0) = 0, y(1) = 0, h = 1/2$

Ques 40.

Solve the wave equation $u_{tt} = u_{xx}$ with the initial and boundary conditions

$u(x, 0) = 0, u_t(x, 0) = 0, u(0, t) = 0, u(1, t) = 200 \sin \pi t$ .

with $h = k = 0.25$ , using the explicit method upto four time levels.

Ques 41.

Find an approximate value of y(1.0) for the initial value problem

$y' = x^3 - y^3, y(0) = 1$

using the multiple method

$y_{n+1} = y_n + \frac{h}{3} [7f_n - 2f_{n-1} + f_{n-2}]$

with step length $h = 0.25$ . Calculate the starting values using Runge-Kutta second order method with the same h.

Ques 42.

Ques 43.

Find an approximate value of y(1.0) for the initial value problem

y = x - 2y, y(0) = 1

using Milne-Simpson’s method

$$y_{n+1} = y_{n-1} + \frac{h}{3} [f_{n+1} + 4f_n + f_{n-1}]$

with the step length $h = 0.2$ . Calculate the starting value using Runge-Kutta fourth order method with the same h.

Ques 44.

) Using fourth order Taylor series method with $h = 0.2$ , solve the initial value problem

$$y' = x + \cos y, \quad y(0) = 0$

upto $x = 1$ .

~~Rs. 200~~
Rs. 123

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IGNOU MSCMACS Assignments Jan - July 2025 - IGNOU University has uploaded its current session Assignment of the MSCMACS Programme for the session year 2026. Students of the MSCMACS Programme can now download Assignment questions from this page. Candidates have to compulsory download those assignments to get a permit of attending the Term End Exam of the IGNOU MSCMACS Programme.

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Course Name	M.Sc. Mathematics with Applications in Computer Science
Course Code	MSCMACS
Programm	MASTER DEGREE PROGRAMMES Courses
Language	English

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