Chap1-2

# Chap1-2 - Engineering Analytical Methods 1 Department of...

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Engineering Analytical Methods 1 Department of Aerospace & Mechanical Engineering University of Southern California Los Angeles, CA 90089-1453 May 19, 2009 1 c s Sadhal 2009. This document is for the exclusive use of the students registered in USC’s AME 526 Course. Any unauthorized copying, downloading or transmission (electronic or otherwise) is forbidden.

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Contents 1B a s i cC o n c e p t s 3 1 . 1 L in e a randN on l e a rEqu a t i s......................... 3 2 Linear Ordinary Di f erential Equations 5 2.1 Homogeneous Equations with Constant Coe c i en t s.............. 5 2.2 Non-Homogeneous Di f e r t i a lEqu a t i s.................... 1 0 2.2.1 The Method of Undetermined Coe c i t s ............... 1 0 2 . 2 . 2 V a r i a t i ono fP a r am e t e r s ......................... 2 0 2 . 3 Th eEu l e a t i on. ............................... 2 3 2 . 3 . 1 H om o g e ou sEu l e a t i on . ..................... 2 5 2 . 3 . 2 N -H o g e l e a t i .................. 2 8 3 Systems of Linear Di f erential Equations 33 3 . o g e sSy s t em s .............................. 3 4 3 . 1 . 1 Sy s t sw i thD i s t c tE i g v a lu e 3 4 3 . 1 . 2 Sy s t i thR ep e a t edE i g v a e s................... 3 9 3 . -h o g e s t s............................ 4 2 3.2.1 The Method of Undetermined Coe c i t 4 2 3 . 2 . 2 M e th odo fV a r i a t i a r e t e r s ................... 4 4 1

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2 CONTENTS
Chapter 1 Basic Concepts 1 1.1 Linear and Nonlinear Equations Ad i f erential equation is a relationship between a function and its derivatives. For many physical systems such relationships usually describe the behavior of the corresponding sys- tems. If f ( x ) is a function of x , then the following relationships can be considered to be di f er- ential equations: d 2 f dx 2 + f df dx +6 f =0 (1 .1) d 2 f dx 2 + x 2 d 2 f dx 2 +3 f (1.2) d 3 f dx 3 + X dx ~ 2 d 2 f dx 2 +4 dx + f .3) d 2 f dx 2 +sin x dx +cos xf (1.4) Here, x is the independent variable and f is the dependent variable. Equations (1.2) and (1.4) are linear equations while (1.1) and (1.3) are considered to be nonlinear. The nonlinearity comes about through the presence of terms involving products of f and its derivatives. For example, X dx ~ 2 d 2 f dx 2 ,f dx are nonlinear terms. However, terms such as: x 2 dx and sin x dx 1 c s Sadhal 2009. This document is for the exclusive use of the students registered in USC’s AME 526 Course 3

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4 BASIC CONCEPTS c s Sadhal do not make an equation nonlinear. Linear di f erential equations have an important property that when di f erent solutions of the equation are added together, the sum also represents a solution. For example, the equation: d 2 y dx 2 +6 dy dx +9 y =0 (1.5) has two solutions: y 1 ( x )=e 3 x and y 2 ( x )= x e 3 x At the same time, any linear combination: y = C 1 e 3 x + C 2 xe 3 x is also a solution.
Chapter 2 Linear Ordinary Di f erential Equations An n -th order linear ordinary di f erential equation may be written in the following general form: a n ( x ) y ( n ) ( x )+ a n 1 ( x ) y ( n 1) ( x ... + a 1 ( x ) y I ( x a 0 ( x ) y ( x )= f ( x ) , (2.1) where the superscript denotes derivatives as shown below: y ( k ) ( x X d dx ~ k y ( x ) .

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Chap1-2 - Engineering Analytical Methods 1 Department of...

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