First_5_files - 1 MATH2860U: Chapter 1 1 INTRODUCTION TO...

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MATH2860U: Chapter 1 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS Definitions and Terminology (Section 1.1, pg. 2) Recall: Back in Calculus II, we introduced the concept of a differential equation. Definition: A differential equation is an equation which contains the derivatives of an unknown function (dependent variable) with respect to one or more independent variables. Notation: In order for us to talk about differential equations and figure out how to solve them, it helps to classify them according to a few criteria: type, order, and linearity. Ordinary Differential Equation vs Partial Differential Equation Definition: An ordinary differential equation ( ODE ) is one in which the unknown function depends on a single independent variable. A partial differential equation ( PDE ) is one in which the unknown function depends on more than one independent variables. Order of a Differential Equation Definition: The order of a differential equation (either ODE or PDE) is the order of the highest derivative in the equation. 1 Differential Equation
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MATH2860U: Chapter 1 2 Note: If we write a differential equation in the form ) , , , , , ( ) 1 ( n n n y y y y x f dx y d where f is a real-valued continuous function, then the differential equation is said to be in normal form . Linear vs Nonlinear Differential Equations Definition: An n th -order differential equation 0 ) , , , , , ( ) ( n y y y y x F is said to be linear if F is linear in ) ( , , , , n y y y y . i.e. any general linear ODE of order n is: ) ( ) ( ) ( ) ( ) ( 0 1 ) 1 ( 1 ) ( x g y x a y x a y x a y x a n n n n Any equation which is not of this form is called a nonlinear equation. Note: Nonlinear DEs are typically much harder to solve than linear DEs; because of this, linearization is a common strategy which is used to obtain a linear problem from a nonlinear one in the hopes that the linear problem is easier to analyze. Systems of Differential Equations Another way to classify differential equations is based on the number of unknown functions that are involved. If a differential equation has more than one unknown (dependent variable), then more than one function is to be determined and hence, more than one equation is required…in this case, we have a system of differential equations. 2 Differential Equation
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