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Unformatted text preview: Ch 2.1: Linear Equations; Method of Integrating Factors A linear first order ODE has the general form where f is linear in y . Examples include equations with constant coefficients, such as those in Chapter 1, or equations with variable coefficients: ) , ( y t f dt dy = ) ( ) ( t g y t p dt dy = + b ay y + = Constant Coefficient Case For a first order linear equation with constant coefficients, recall that we can use methods of calculus to solve: C at e k ke a b y C t a a b y dt a a b y dy a a b y dt dy = + = + = = = , / / ln / / / , b ay y + = Variable Coefficient Case: Method of Integrating Factors We next consider linear first order ODEs with variable coefficients: The method of integrating factors involves multiplying this equation by a function ( t ), chosen so that the resulting equation is easily integrated. ) ( ) ( t g y t p dt dy = + Example 1: Integrating Factor (1 of 2) Consider the following equation: Multiplying both sides by ( t ), we obtain We will choose ( t ) so that left side is derivative of known quantity. Consider the following, and recall product rule: Choose ( t ) so that 2 / 2 t e y y = + [ ] y dt t d dt dy t y t dt d ) ( ) ( ) ( + = t e t t t 2 ) ( ) ( 2 ) ( = = ) ( ) ( 2 ) ( 2 / t e y t dt dy t t = + Example 1:...
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 Fall '10
 BenjaminWalter

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