tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 1

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 3 part 1', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | y 3xy 2y x2 y y y The degree of a differential equation is the highest power of the highest derivative in the equation. The following equations are first second and third degree respectively. yt 3y2 sin x y z 2 2x cos y ex y 3 y5 0 An equation is said to be linear if it is linear in the dependent variable. y cos x x2y 0 is a linear differential equation. y xy2 0 is a nonlinear differential equation. A differential equation is homogeneous if it has no terms that are functions of the independent variable alone. Thus an inhomogeneous equation is one in which there are terms that are functions of the independent variables alone. y xy y 0 is a homogeneous equation. y y x2 0 is an inhomogeneous equation. A first order differential equation may be written in terms of differentials. Recall that for the function y x the differential dy is defined dy y x dx. Thus the differential equations y x2y and y xy2 sin x can be denoted dy x2y dx and dy xy2 dx sin x dx. 774 A solution of a differential equation is a function which when substituted into the equation yields an identity. For example y x In x is a solution of y -y 1. x We verify this by substituting it into the differential equation. In x 1 In x 1 We can also verify that y cex is a solution of y y 0 for any value of the parameter c. c ex c ex 0 Example Problems In this section we will discuss physical and geometrical problems that lead to first order differential equations. Growth and Decay Example Consider a culture of bacteria in which each bacterium divides once per hour. Let n t E N denote the population let t denote the time in hours and let n0 be the population at time t 0. The population doubles every hour. Thus for integer t the population is n02 . Figure shows two possible populations when there is initially a single bacterium. In the first plot each of the bacteria divide at times t m for m E N. In the second plot they divide at times t m 1 2. For both plots the population is 2 for .