B.Sc. Mathematics Honours GE-4 Question Paper 2019 (CBCS)

1. Answer any five questions: 2x5
(a) What are the sources of errors in numerical computation?
(b) Write down the sufficient condition for the convergence of the Gauss - Seidel - iteration method.
(c) Write the advantages and disadvantages of fixed point iteration method.
(d) Prove that $\Delta\nabla f(x) = \Delta f(x) - \nabla f(x)$, where the symbols $\Delta$ and $\nabla$ carry their usual meaning.
(e) Write the formula of Runge - Kutta method of order four to solve the initial value problem $y' = f(x,y)$ with $y(x_{0}) = y_{0}$.
(f) Define 'Degree of precision' of a numerical integration formulae.
(g) Why relative error is a better indicator of the accuracy of a computation than the absolute error?
(h) Show by an example that the Simpson's $\frac{1}{3}$ rule is exact for integrating a polynomial of degree 3.

2. Answer any four questions: 5x4=20
(a) Describe Newton Raphson method for computing a simple root of an equation $f(x) = 0$. What is the sufficient condition for convergent of Newton - Raphson method?
(b) Derive the Simpson's one third integration formula in the form
$$\int_{a}^{b} f(x)dx = \frac{b-a}{6} [f(a) + 4f(\frac{a+b}{2}) + f(b)] - \frac{(b-a)^{5}}{2^{5} \times 90} f^{iv}(z)$$
where $a < z < b$.
(c) Solve the following system of equation by Gauss - Seidal method correct to three significant figures:
$3x + y + z = 3$
$2x + y + 5z = 5$
$x + 4y + z = 2$
(d) Apply Runge Kutta method of order 4, find the values of $y$ at 0.1 where $y' = x^{2} + y^{2}$ with $x = 0, y = 1$.
(e) Show that the n-th order divided difference of a polynomial of degree n is constant.

3. Answer any one question: [1x10]
(a) (i) Establish Newton's forward difference interpolation formula for the equispaced interpolating points.
(ii) Construct Lagrange's interpolation polynomial for the function $y = \sin \pi x$, choosing the points $x_{0} = 0, x_{1} = \frac{1}{6}, x_{2} = \frac{1}{2}$.
(b) (i) Establish Newton - cotes quadrature formula for numerical integration of $f(x)$ in [a, b] whose functional values are unknown at (n + 1) equispaced distinct points.
(ii) Write down the modified Euler formula to solve the differential equation $\frac{dy}{dx} = f(x,y), y(x_{0}) = y_{0}$ and state why it is better than Euler method.


1. Answer any ten questions out of following fifteen questions: [10x2]
(a) What is the order and degree of the following partial differential equation:
$(\frac{\partial z}{\partial x})^{2} + \frac{\partial^{3}z}{\partial y^{3}} = 2x(\frac{\partial z}{\partial x})$
(b) Eliminate arbitrary constants a and b from $z = (x-a)^{2} + (y-b)^{2}$ to form the partial differential equation.
(c) Show that characteristics equation of the partial differential equation $x^{2}r + 2xys + y^{2}t = 0$ represents a family of straight lines passing through origin.
(d) Define semi-linear and Quasi linear partial differential equation.
(e) The equation $\frac{\partial^{2}u}{\partial t^{2}} = c^{2} \frac{\partial^{2}u}{\partial x^{2}}$ is (i) Parabolic, (ii) hyperbolic, (iii) elliptic, (iv) none of these.
(f) Find the complete integral of $yp + xq = pq$.
(g) Solve: $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = xyz$.
(h) Classify heat equation and Laplace equation.
(i) Classify the following partial differential equation: $\frac{\partial^{2}u}{\partial x^{2}} + 4(\frac{\partial^{2}u}{\partial x\partial y}) + 4 \frac{\partial^{2}u}{\partial y^{2}} = 0$.
(j) Find complete integral of $z = px + qy + p^{2} + q^{2}$.
(k) Write down D'Alemberts formula for the non-homogeneous wave equation.
(l) Eliminate arbitrary functions f and F from $y = f(x-at) + F(x+at)$ to form the partial differential equation.
(m) State basic existence theorem for Cauchy problem.
(n) Write the heat conduction and Laplace equation.
(o) If the Partial differential equation $(x-1)^{2}u_{xx} - (y-2)^{2}u_{yy} + 2xu_{x} + 2yu_{y} + 2xyu = 0$ is parabolic in $S \subseteq R^{2}$ but not in $R^{2} \setminus S$ then S is: (i) $\{(x,y) \in R^{2}: x=0 \text{ or } y=2\}$, (ii) $\{(x,y) \in R^{2}: x=1 \text{ and } y=2\}$, (iii) $\{(x,y) \in R^{2}: x=1\}$, (iv) $\{(x,y) \in R^{2}: y=2\}$.

2. Answer any four questions out of six questions: [5x4]
(a) Using Lagrange's method solve the partial differential equation $z(x+y)p + z(x-y)q = x^{2} + y^{2}$.
(b) Write down the canonical form of one-dimensional wave equation: $\frac{\partial^{2}z}{\partial x^{2}} - \frac{\partial^{2}z}{\partial y^{2}} = 0$.
(c) Find the integral surface of the linear partial differential equation $x(y^{2}+z)p - y(x^{2}+z)q = (x^{2}-y^{2})z$ which contains the straight line $x+y=0, z=1$.
(d) A particle moves in the curve $y = a \log_{e} \sec(\frac{x}{a})$ in such a way that the tangent to the curve rotates uniformly. Prove that the resultant acceleration of the particle varies as the square of curvature.
(e) Establish the formula: $\frac{d^{2}u}{d\theta^{2}} + u = \frac{P}{h^{2}u^{2}}$ for the motion of a particle describing a central orbit under an attractive force P per unit mass.
(f) Obtain the PDE which has its general solution $z = xf(\frac{y}{x})$, where f is an arbitrary function.

3. Answer any two questions out of four questions: [10x2]
(a) Use the method of separation of variables to determine the solution $u(x,y)$ of the Laplace equation $\frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}u}{\partial y^{2}} = 0$ with boundary conditions: $u(x,0) = f(x), u(x,\pi) = 0, u(0,y) = u(\pi,y) = 0$ for $0 \le x, y \le \pi$.
(b) Find the solution of the initial boundary value problem $u_{tt} = u_{xx}, 0 < x < 2, t > 0$ with $u(x,0) = \sin(\frac{\pi x}{2}), u_{t}(x,0) = 0, u(0,t) = 0, u(2,t) = 0$.
(c) Reduce the equation $\frac{\partial^{2}z}{\partial x^{2}} + 2 \frac{\partial^{2}z}{\partial x\partial y} + \frac{\partial^{2}z}{\partial y^{2}} = 0$ to canonical form and hence solve it.
(d) Find the solution of one-dimensional diffusion equation $k \frac{\partial^{2}u}{\partial x^{2}} = \frac{\partial u}{\partial t}$ satisfying: (i) u is bounded as $t \rightarrow \infty$, (ii) $u_{x}(0,t) = 0, u_{x}(a,t) = 0$, (iii) $u(x,0) = x(a-x), 0 < x < a$.