1.1 Definitions and Terminology of Differential Equations 1. Differential Equations: An ordinary differential equation (ODE) is an equation that contains the derivatives of one or more dependent variables with respect to a single independent variable. For example, d2y dy y! " 5y # ex, " x2 ! 4y # ln!x" dx2 dx A partial differential equation (PDE) is an equation that contains the derivatives of one or more dependent variables with respect to two or more independent variables. For example, 2 2 "u " "u # ex"y, " u " " u # ln!x" "x "y "x2 dy2 Order of a Differential Equation: The order of a differential equation is the order of the highest derivative in the equation. In this course, we mainly study the ordinary differential equation. A general form of an ODE of order n : F x, y, y!,...,y!n" # 0 For example, y! " 5y ! ex # 0 Linear and Nonlinear Differential Equations: A linear differential equation contains only: y, y!, y!!,.... Forexample, dy 2 1 ! y y! " y # 0, " y # ex ! " dx are nonlinear differential equations. 2. Solution of an Ordinary Differential Equation: A solution of an nth-order ODE F!x,y,y!,...,y!n" " # 0 is a function y!x" which satisfies the differential equation for x in an interval I. The interval I is said to be the interval of definition (or the domain of the solution). The solution is called a trivial solution if it is a zero solution. Explicit solution: y # f!x" Implicit solution: G!x,y" # 0 Example Show that the function y # C1 sin!!x" " C2 cos!!x" where C1 and C2 are constants is a solution of the differential equation: y!! " !2y # 0. Show that y satisfies the differential equation: ! y # C1! cos!!x" ! C2! sin!!x" !! 2 2 2 y # !C1! sin!!x" ! C2! cos!!x" # !! !C1 sin!!x" " C2 cos!!x"" !! 2 2 2 y " ! y # !! !C1 sin!!x" " C2 cos!!x"" " ! !C1 sin!!x" " C2 cos!!x"" # 0 1 In this solution, constants C1 and C2 are arbitrary. Solution y # C1 sin!!x" " C2 cos!!x" is called a two-parameter family of solution and also called the general solution of the differential equation y!! " !2y # 0. There are infinitely many solutions in this family: C1 # 2, C2 # 0:y # 2sin!!x" C1 # 0, C2 # 2:y # 2cos!!x" C1 # 2, C2 # !1:y # 2sin!!x" ! cos!!x" 2 2 2 1 1 1 0 0 0 -2 -1x 1 2 -2 -1 1x 2 -2 -1x 1 2 -1 -1 -1 -2 -2 -2 y # 2sin!!x" y # 2cos!!x" y # 2sin!!x" ! cos!!x" 2 2 C1 C2 y # C1 sin!!x" " C2 cos!!x" # C1 " C2 sin!!x" " cos!!x" 2 2 2 2 C1 " C2 C1 " C2 Let C C cos!"" # 1 , sin!"" # 2 . 2 2 2 2 C1 " C2 C1 " C2 Then C C tan " # 2 and " # arctan 2 C1 C1 2 2 C2 y # C1 " C2 sin !x " arctan C1 C1 and C2 change the magnitude and shift the sine wave to the left or to the right. Example Show that the equation !2x2y " y2 # 1 is a solution of the differential equation 2xydx " !x2 ! y"dy # 0. Show the equation ! 2x2y " y2 # 1 satisfies the differential equation: dy dy dy d !2x2y " y2 # d 1 $ !4xy ! 2x2 " 2y # 0 $ !2 x2 ! y # 4xy dx ! " dx ! " dx dx ! " dx ! !x2 ! y"dy # 2xydx $ 2xydx " !x2 ! y"dy # 0 Solve the equation !2x2y " y2 # 1: y2 ! 2x2y ! 1 # 0 2x2 % 4x2 ! 4!!1" y # # x2 % x2 " 1 2 y # x2 " x2 " 1 , y # x2 ! x2 " 1 are two explicit solutions. 2 Example Find the value of m so that y # xm is a solution of the differential equation x2y!! ! 7xy! " 15y # 0. y! # mxm!1, y!! # m!m ! 1"xm!2, x2y!! ! 7xy! " 15y # x2!m"!m ! 1"xm!2 ! 7xmxm!1 " 15xm # xm!m!m ! 1" ! 7m " 15" # 0 m2 ! m ! 7m " 15 # m2 ! 8m " 15 # !m ! 3"!m ! 5" # 0, m # 3orm # 5. Solutions: y # x3, y # x5. 3.