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# Math 250B 1.2 Math 250B

Cal State Fullerton

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This 4 page Class Notes was uploaded by Sharlyn Dee on Sunday February 7, 2016. The Class Notes belongs to Math 250B at California State University - Fullerton taught by in Winter 2016. Since its upload, it has received 22 views. For similar materials see Math 250B Intro Linear Algebra and Differential Equations in Math at California State University - Fullerton.

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Date Created: 02/07/16

Section 1.2: Basic Ideas and Terminology Definition 1.2.1: A differential equation is an equation involving one or more derivatives of an unknown function. To begin our study of differential equation we need some common terminology. If an equation involves the derivative of one variable with respect with another, then the former is called a dependent variable and the later an independent variable. Example 1: d x dx a kx 0 dt2 dt A differential equation involving ordinary derivatives with respect to a single independent variable is called an ordinary differential equation. A differential equation involving partial derivatives with respect to more than one independent variable is a partial differential equation. Example 2: Classify each of the following as ordinary or partial differential equation. d x dx (a) 2 a kx 0 dt dt 2 2 (b) d u a d u 0 dx 2 dy 2 Definition 1.2.3: The order of the highest derivative occurring in a differential equation is called the order of the differential equation. Definition 1.2.4: A differential equation that can be written in the form: a x a x n1 a x F x 0 1 n where a , ,a and F are functions of x only, is called a linear differential equation of order n. Such 0 n a differential equation is linear in y,y',y'', ,y Example3: Determine the order of the given differential equation and state whether is linear or non-linear. 2 (a) d y y 0 dx 2 Page 1 of 4 d y 3 (b) 4 y x dx d y dy (c) 3 y cos x dx dx 2 2 (d) xy 4x y 2 y 0 1 x (e) y x y y 02 Page 2 of 4 Definition 1.2.7: A function y f that is (at least) n times differentiable on an interval I is called a solution to the differential equation on I if the substitution y f , y' f , , y f reduces the differential equation to an identity valid for all x in I. In this case we say that y f satisfies the differential equation. Definition 1.2.11: A solution to an nth-order differential equation on an interval I is called the general solution on I if it satisfies the following conditions: 1. The solution contains n constants c1,c2, ,cn 2. All solutions to the differential equation can be obtained by assigning appropriate values to the constants. Example 4: Page 3 of 4 Definition 1.2.13: An nth order differential equation together with n auxiliary conditions of the form n y 0 y 0 y ' x 0 y 1 , y 0 y 1n where y 0y 1 ,yn1 are constant, is called an initial-value problem. Theorem 1.2.15: Let a1,a2, ,an,F be functions that are continuous on an interval I. Then, for any x0 in I, the initial-value problem n y a 1 a n1 'a xn F x n y 0 y 0 y ' x 0 y 1 , y 0 y 1n Example 5: Page 4 of 4

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