Math118_S10_W4

Math118_S10_W4 - Math118 (M. Kohandel) -Outline (week 4)...

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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 1 Outline (week 4) 4.1 Introduction (15.1) Definitions Classifications 4.2 Separable differential equation (15.2) 4.3 Linear first-order differential equation (15.3)
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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 2 4.1 Introduction We saw that derivative dx dy / of a function ) ( x f y is itself another function of x found by some appropriate rules, e.g. xy dx dy xe dx dy e y x x 2 2 2 2 Here x is the independent variable, and y is the dependent variable. The problem we face in this week is if we are given a differential equation, can we find the unknown function ) ( x f y ? Definition: An equation containing the derivatives of dependent variable with respect to a single independent variable, is called an ordinary differential equation (ODE), e.g. 0 3 , 2 2 2 y dx dy dx y d e x dt dx t Differential equations serve as models for many problems in engineering and science. Physical quantities are represented by functions, and laws of physics relate these functions to their rates of changes (derivatives); which results in DEs. Terminology: ODEs can be classified by orders and linearity. The order of a ODE is the order of the highest derivative in the equation, e.g. ODE osrder 2nd 0 ) ( 2 4 3 2 2 y dx dy dx y d A ODE is said to be linear if only contains linear functions of y and its derivatives. e.g. ODE) nonlinear ( sin 2 ) 1 ( , ODE) linear ( 0 2 4 2 2 x y y y x dx dy x dx y d Particular and general solutions: A particular solution of a ODE is any function which satisfies the ODE (contains no arbitrary constant). The general solution is the set of all possible solutions . Example 4.1: Consider the DE 0 4 y y . The functions ) 2 cos( x y and ) 2 sin( 3 x y satisfy the equation – these are particular solutions. ) 2 sin( ) 2 cos( 2 1 x c x c y is the general solution. We need extra information to obtain the constants.
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Math118_S10_W4 - Math118 (M. Kohandel) -Outline (week 4)...

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