# ch02_1 - Ch 2.1 Linear Equations Method of Integrating...

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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 μ ( 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 μ ( t ), we obtain We will choose μ ( t ) so that left side is derivative of known quantity. Consider the following, and recall product rule: Choose μ ( t ) so that 2 / 2 t e y y = + ′ [ ] y dt t d dt dy t y t dt d ) ( ) ( ) ( μ μ μ + = t e t t t 2 ) ( ) ( 2 ) ( = ⇒ = ′ μ μ μ ) ( ) ( 2 ) ( 2 / t e y t dt dy t t μ μ μ = + Example 1:...
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ch02_1 - Ch 2.1 Linear Equations Method of Integrating...

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