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MMT 7: Differential Equations and Numerical Solutions

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 7/Assignment-1/2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMTE 007 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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MMT 7 2025 - English

Course Code: MMT-007

Assignment Code: MMT-007/TMA/2025

Maximum Marks: 100

1. a) Solve the differential equation:

x²y" +6xy'+(6+x²)y=0

in series about x = 0.

b) Express f(x)=x⁴+ 3x³+ 4x²- x + 2 in terms of Legendre polynomials.

2. a) Using method of Ferobenius, find the solution of the differential equation:

$x^2frac{d^2 y}{dx^2}+(x+x^2)frac{dy}{dx}+(x-9)y=0$

near x = 0.

b) Find:

$L^{-1}left{frac{s}{(s^2+4)^2} ight}$

c) Find L{F(t)), if:

$F(t)=egin{cases}sinleft(t-frac{pi}{4} ight),&t>frac{pi}{4}\0,&t<frac{pi}{4}end{cases}$

3. a) Find the Fourier transform of $e^{-9x^2}$

b) Find the solution of the heat conduction equation subject to the given initial and boundary conditions:

$frac{partial u}{partial t}=frac{partial^2 u}{partial x^2},quad 0le xle 1$

$u(x,0)=sin(pi x)quad ext{for}0le xle 1$

$u(0,t)=0=u(1,t)$

Using Laasonen method with $lambda=frac{1}{6}$ and . $h=frac{1}{3}$ Integrate for two levels.

4. a) Find the solution of the initial boundary value problem:

$frac{partial u}{partial t}=frac{partial^2 u}{partial x^2}$

subject to given initial and boundary conditions:

$u(x,0)=2xquad ext{for}xinleft[0,frac{1}{2} ight]$

$ext{and},,2(1-x) ext{for}left[frac{1}{2},1 ight]$

$u(0,t)=0=u(1,t)$

You may use step length along x-axis, h = 2.0 and solve by Schmidt method with mesh ratio $lambda=frac{1}{6}$

b) Show that the method.

$y_{i+1}=frac{4}{3}y_i-frac{1}{3}y_{i-1}+frac{2h}{3}y'_{i+1}$

is A-stable when applied to test equation y′ = λy, λ < 0 .

5. a) Use Fourier transforms to solve the boundary value problem:

$frac{partial u}{partial t}=4frac{partial^2 u}{partial x^2},quad-infty<x<infty,quad t>0$

subject to the conditions:

i) $u,frac{partial u}{partial x} o 0quad ext{as}quad x opminfty$

ii) $u(x,0)=f(x)$

b) Solve the initial value problem y' = x² + y²y(0) = 1, upto x = 2.0 using third order Taylor series method with h = 0.1.

6. a) Using Laplace transform, solve the equation:

$frac{partial^2 u}{partial t^2}=9frac{partial^2 u}{partial x^2}$

given that:

$u(0,t)=u(2,t)=0,quad u_t(x,0)=0$

$ext{and}u(x,0)=10sin(2pi x)-20sin(5pi x)$

b) Using second order finite difference method, solve the boundary value problem:

$frac{d^2 y}{dx^2}=frac{3}{2}y^2quad ext{with}y(0)=4,y(1)=1$

Using step length $h=frac{1}{3}$

7. Find the solution of $abla^2 u$ = 0 in R subject to the boundary conditions:

$u(x,y)=x^2-y^2quad ext{on}x=0,y=0,y=1;$

$u+frac{partial u}{partial x}=x^2+2x-y^2quad ext{on}x=1,$

where R is the square 0≤ x ≤1,0≤ y ≤1, using the five point formula. Use central difference approximation in the boundary condition. Assume uniform step length h = 1/2 along the axes.

8. a) Use finite Fourier transform to solve:

$frac{partial u}{partial t}=kfrac{partial^2 u}{partial x^2},quad 0<x<4,quad t>0$

Subject to the conditions:

u(x,0) = 2x, 0 < x < 4

and u(0,t) = u(4, t) = 0

b) Solve the boundary value problem:

y"+y+f(x)=0

y'(0) = 0, y(1) = 0

by determining the appropriate Green’s function by using the method of variation of parameters and expressing the solution as a definite intergral.

9. Solve the boundary value problem:

y′′ − 3y′ + 2y = 2

with

y(0) - y'(0) = -1

y(1) + y'(1) = 1

using the second order finite difference method with step length $h=frac{1}{2}$

10. a) Using the generating function J_n(x), prove that J_n-1(x)+J_n+1(x)= $frac{2n}{x}J_n$ (x), for integer values X of n.

b) Evaluate:

$int_{-1}^{1}frac{P_n(x)}{sqrt{1-2xt+t^2}},dx$

where P_n(x) is Legendre polynomial.

MMT 7 2026 - English

Assignment (MMT-007)

Course Code: MMT-007
Assignment Code: MMT-007/TMA/2026
Maximum Marks: 100

1. a) 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.

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

2. a) 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) $(3x + 1)xy'' - (x + 1)y' + 2y = 0$

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

c) 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:

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

b) 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.

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

4. a) Find the Laplace transform of $\frac{\sin \sqrt{t}}{\sqrt{t}} \dots$

b) 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}$

5. a) 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.$

b) 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}}}.$

6. a) 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), $-\infty < x < \infty$ .

b) 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}$

7. a) 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.

b) 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:

8. a) Using second order finite Difference method, solve the boundary value problem

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

b) 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.

9. a) 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.

b) 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 $h = k = 1/2$ .

10. a) 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.

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

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

upto $x = 1$ .

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