Laplace, transformadas , inversas, etc

MATHEMATICS-I

LAPLACE TRANSFORMS
I YEAR B.Tech

By Y. Prabhaker Reddy
Asst. Professor of Mathematics Guru Nanak Engineering College Ibrahimpatnam, Hyderabad.

SYLLABUS OF MATHEMATICS-I (AS PER JNTU HYD)
Name of the Unit 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Name of the Topic Basic definition of sequences and series Convergence and divergence. Ratio test Comparison test Integral test Cauchy’sroot test Raabe’s test Absolute and conditional convergence

Unit-I Sequences and Series

Unit-II Functions of single variable

2.1 Rolle’s theorem 2.2 Lagrange’s Mean value theorem 2.3 Cauchy’s Mean value theorem 2.4 Generalized mean value theorems 2.5 Functions of several variables 2.6 Functional dependence, Jacobian 2.7 Maxima and minima of function of two variables 3.1 Radius , centreand Circle of curvature 3.2 Evolutes and Envelopes 3.3 Curve Tracing-Cartesian Co-ordinates 3.4 Curve Tracing-Polar Co-ordinates 3.5 Curve Tracing-Parametric Curves 4.1 Riemann Sum 4.3 Integral representation for lengths 4.4 Integral representation for Areas 4.5 Integral representation for Volumes 4.6 Surface areas in Cartesian and Polar co-ordinates 4.7 Multiple integrals-double and triple 4.8Change of order of integration 4.9 Change of variable 5.1 Overview of differential equations 5.2 Exact and non exact differential equations 5.3 Linear differential equations 5.4 Bernoulli D.E 5.5 Newton’s Law of cooling 5.6 Law of Natural growth and decay 5.7 Orthogonal trajectories and applications 6.1 Linear D.E of second and higher order with constant coefficients 6.2 R.H.S term of the formexp(ax) 6.3 R.H.S term of the form sin ax and cos ax 6.4 R.H.S term of the form exp(ax) v(x) 6.5 R.H.S term of the form exp(ax) v(x) 6.6 Method of variation of parameters 6.7 Applications on bending of beams, Electrical circuits and simple harmonic motion 7.1 LT of standard functions 7.2 Inverse LT –first shifting property 7.3 Transformations of derivatives and integrals 7.4 Unit step function, Secondshifting theorem 7.5 Convolution theorem-periodic function 7.6 Differentiation and integration of transforms 7.7 Application of laplace transforms to ODE 8.1 Gradient, Divergence, curl 8.2 Laplacian and second order operators 8.3 Line, surface , volume integrals 8.4 Green’s Theorem and applications 8.5 Gauss Divergence Theorem and applications 8.6 Stoke’s Theorem and applications

Unit-IIIApplication of single variables

Unit-IV Integration and its applications

Unit-V Differential equations of first order and their applications

Unit-VI Higher order Linear D.E and their applications

Unit-VII Laplace Transformations

Unit-VIII Vector Calculus

CONTENTS
UNIT-7

LAPLACE TRANSFORMS
 Laplace Transforms of standard functions  Inverse LT- First shifting Property Transformations of derivatives and integrals  Unit step function, second shifting theorem  Convolution theorem – Periodic function  Differentiation and Integration of transforms  Application of Laplace Transforms to ODE

LAPLACE TRANSFORMATION
INTRODUCTION Laplace Transformations were introduced by Pierre Simmon Marquis De Laplace (1749-1827), a French Mathematician known as a Newton of French.Laplace Transformations is a powerful Technique; it replaces operations of calculus by operations of Algebra. Suppose an Ordinary (or) Partial Differential Equation together with Initial conditions is reduced to a problem of solving an Algebraic Equation. Definition of Laplace Transformation: Let the Laplace Transformation of Here, is defined as is known as determining ) is called be a given functiondefined for all , then

is called Laplace Transform Operator. The function

function, depends on . The new function which is to be determined (i.e. F generating function, depends on . Here NOTE: Here Question will be in and Answer will be in . Laplace Transformation is useful since  Particular Solution is obtained without first determining the general solution

 Non-Homogeneous...