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Lectur15 - Lecture XV Review of Dynamic Mathematics II I...

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Lecture XV Review of Dynamic Mathematics II I. Examples A. Homogeneous First Order Linear Differential Equations 1. Example 1: x x + = 3 0 Proposed solution x t Ae t ( ) = - 3 Given an initial value of x(0) = 5, what is the value of A? Check the solution: ( 29 x t e x t e x t x t e e t t t t ( ) ’( ) ’( ) ( ) = = - + = - + = - - - - 5 15 3 15 3 5 0 3 3 3 3 2. Example 2: x x + = 6 0 Proposed solution x t Ae t ( ) = - 6 Given the initial value of x(0)=4, what is the value of A? Check the solution: ( 29 x t e x t e x t x t e e t t t t ( ) ’( ) ’( ) ( ) = = - + = - + - - - - 4 24 24 6 4 6 6 6 6 B. Non Homogeneous First Order Linear Differential Equations with a Constant Right Hand Side 1. Example 1: x x + = 3 6 Proposed solution x t x e t ( ) = - + - 0 3 6 3 6 3 What is the particular solution to this problem? What is the complementary solution to this problem? Given an initial condition of x(0)=5, check the solution: ( 29 ( 29 x t e x t e x t e x t x t e e t t t t t ( ) ( ) ’( ) ’( ) ( ) = - + = + = - + = - + + = - - - - - 5 2 2 3 2 9 3 9 3 3 2 6 3 3 3 3 3 2. Could you solve for the initial conditions, if you where given x(2) = 10?

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( 29 x x e x e ( ) ( ) 2 2 2 10 10 2 2 0 4 0 4 = - + = = - + - 3. Example 2: x x + = 6 6 Proposed solution x t x e t ( ) = - + - 0 6 6 6 6 6 Taking the initial solution of x(0)=4 and checking the solution: ( 29 x t e x t e x t x t e e t t t t ( ) ’( ) ’( ) ( ) = + = - + = - + + = - - - - 3 1 18 6 18 6 3 1 6 6 6 6 6 C.
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