Note for Mathematics-3 - M-3 By LOKESH SAHU

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Subject: Mathematics III Subject Code: BSCM1205 Branch: B. Tech. all branches Semester: (3rd SEM) Lecture notes prepared by: i) Dr. G. Pradhan (Coordinator) Asst. Prof. in Mathematics College of Engineering and Technology, Bhubaneswar, BPUT ii) Ms. Swagatika Das Lecturer in Mathematics College of Engineering and Technology, Bhubaneswar BPUT iii) Ms. Mita Sharma Lecturer in Mathematics College of Engineering and Technology, Bhubaneswar BPUT iv) Mr. N. C. Ojha Lecturer in Mathematics College of Engineering and Technology, Bhubaneswar BPUT iii) Mr. A. P. Baitharu Lecturer in Mathematics College of Engineering and Technology, Bhubaneswar BPUT iii) Mr. Susant Rout Lecturer in Mathematics College of Engineering and Technology, Bhubaneswar BPUT Disclaimer: The lecture notes have been prepared by referring to many books and notes prepared by the teachers. This document does not claim any originality and cannot be used as a substitute for prescribed textbooks. The information presented here is merely a collection of materials by the committee members of the subject. This is just an additional tool for the teaching-learning process. The teachers, who teach in the class room, generally prepare lecture notes to give direction to the class. These notes are just a digital format of the same. These notes do not claim to be original and cannot be taken as a text book. These notes have been prepared to help the students of BPUT in their preparation for the examination. This is going to give them a broad idea about the curriculum. The ownership of the information lies with the respective authors or institutions. Further, this document is not intended to be used for commercial purpose and the committee faculty members are not accountable for any issues, legal or otherwise, arising out of use of this document. The committee faculty members make no representations or warranties with respect to the accuracy or completeness of the contents of this document and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose.

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BSCM1205 Mathematics-III Module-I (18 hours) Partial differential equation of first order, Linear partial differential equation, Non-linear partial differential equation, Homogenous and non-homogeneous partial differential equation with constant co-efficient, Cauchy type, Monge’s method, Second order partial differential equation The vibrating string, the wave equation and its solution, the heat equation and its solution, Two dimensional wave equation and its solution, Laplace equation in polar, cylindrical and spherical coordinates, potential. Module-II (12 hours) Complex Analysis: Analytic function, Cauchy-Riemann equations, Laplace equation, conformal mapping, Complex integration: Line integral in the complex plane, Cauchy’s integral theorem, Cauchy’s integral formula, Derivatives of analytic functions Module –III (10 hours) Power Series, Taylor’s series, Laurent’s series, Singularities and zeros, Residue integration method, evaluation of real integrals Contents Sl No 1.1 1.2 1.3 1.3 1.4 1.5 1.6 1.7 2.1 2.2 Topics Formation of Partial Differential Equations Linear partial differential equations of First Order Non Linear P.D.Es of first order Charpit’s Method Homogenous partial differential Equations with constant coefficients Non Homogenous partial differential Equations Cauchy type Differential Equation Monge's Method One Dimensional wave equation D Alemberts Solution of wave equation Page No 2

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2.3 2.4 2.5 2.6 2.7 2.8 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 4.2 4.3 4.4 4.5 4.6 Heat Equation Two Dimensional wave equation Laplacian in polar coordinates Circular Membrane ( Use of Fourier-Bessel Series) Laplace’s Equation in cylindrical and Spherical coordinates Solution of partial Differential equation by laplace Transform. Analytic function Cauchy –Reiman equation & Laplace equation Conformal mapping Line integral in complex plane Cauchy’s integral theorem Cauchy’s integral formula Derivatives of analytic function Power Series Taylor series Laurent Series Singularities, Pole, and Residue Residue Integral Evaluation of real integral Formation of Partial Differential Equations In practice, there are two methods to form a partial differential equation. (i) By the elimination of arbitrary constants. (ii) By the elimination of arbitrary functions. (i) Formation of Partial Differential Equations by the elimination of arbitrary constants method: * Let f(x,y,z,a,b)=0 be an equation containing 2 arbitrary constants ''a'' and ''b''. 3

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* To eliminate 2 constants we require at least 3 equations hence we partially differentiate the above equation with respect to (w.r.t) ‘x’ and w.r.t ‘y’ to obtain 2 more equations. * From the three equations we can eliminate the constants ''a'' and ''b''. NOTE 1: If the number of arbitrary constants to be eliminated is equal to the number independent variables, elimination of constants gives a first order partial differential equation. But if the number of arbitrary constants to be eliminated is greater than the number of independent variables, then elimination of constants gives a second or higher order partial differential equation. NOTE 2: In this chapter we use the following notations: p = ∂z/∂x, q = ∂z/∂y, r = ∂2z/∂x2, s = ∂2z/(∂x∂y), t = ∂2z/∂y2 METHOD TO SOLVE PROBLEMS: Step 1: Differentiate the given question first w.r.t ‘x’ and then w.r.t ‘y’. Step 2: We know p = ∂z/∂x and q = ∂z/∂y. Step 3: Now find out a and b values in terms of p and q. Step 4: Substitute these values in the given equation. Step 5: Hence the final equation is in terms of p and q and free of arbitrary constants ''a'' and ''b'' which is the required partial differential equation. (ii) Formation of Partial Differential Equations by the elimination of arbitrary functions method: * Here it is the arbitrary function that gets eliminated instead of the arbitrary constants ''a'' and ''b''. NOTE: The elimination of 1 arbitrary function from a given partial differential equation gives a first order partial differential equation while the elimination of the 2 arbitrary functions from a given relation gives second or higher order partial differential equations. 4