ch02_1 - Ch 2.1: Linear Equations; Method of Integrating...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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:...
View Full Document

Page1 / 16

ch02_1 - Ch 2.1: Linear Equations; Method of Integrating...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online