Solve your IGNOU Doubts

Question:

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

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

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Question:

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:

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Question:

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.

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Question:

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}$

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Question:

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

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Question:

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}}}.$

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Question:

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.$

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Question:

) 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}$

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Question:

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

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Question:

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

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Question:

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.

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Question:

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

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Question:

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:

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Question:

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

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Question:

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

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Question:

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

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Question:

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.

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Question:

  Show the minimum heap after each of the following operations
          i)    Create with values 13, 46, 68, 15, 87, 4, 33, 21, 1, 79.
          ii)   Insertion of value 9.
          iii) Update of value 1 by 71.

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Question:

Construct the Huffman code tree for the set of frequencies in the table below: