Ordinary versus Partial

A Few Notes... I In Algebra, y = f (x) defines the variable y as a function of the variable x. As the value of x changes, so too does the value of y . I In Calculus, the notation dy dx is the derivative of y with respect to x, which is just a fancy way of ‘at a particular x value, the value of y is increasing or decreasing by this rate’. Thus, we will refer to dy dx as the ‘rate of change’ (of y with respect to x). I In Calculus 3 it is possible for y to depend on more than one variable, and we could write y = f (x, t) to denote that y depends on x and t, for example. In this case, we write ∂y ∂x and ∂y to denote the rates of change of y respect to x or t, ∂t respectively. The change in notation is purely so that we remember that y is a function of more than one variable (multivariate function). These are called partial derivatives and they follow pretty much the same rules as regular Calculus 1 derivatives. Definition A differential equation is an equation that involves the rate of change of some dependent variables with respect to one or more of their independent variables. y +1 dy = dx x + xy ∂2y ∂2y + =C ∂x 2 ∂t 2 In the case that the dependent variables depend on only a single independent variable, we say that it is an ordinary differential equation, or ODE for short. Otherwise, if the dependent variables depend on more than one independent variable, we say that it is a partial differential equation, or PDE for short. Example Consider the generic functions f (x), g (x), h(x, y ), and k(x, y ). Identify whether the following differential equations are ordinary or partial. 1. 2. 3. 4. df dx df dx ∂h ∂y dk dx =f +x (ordinary) dg dx (ordinary) + = f · g + x · (f + g ) =1 (partial) =1 (incorrect) This Class... As you can probably tell from the title of this class, we will only be solving ODEs in this class. Therefore, you may put PDEs out of your mind, however do not forget partial derivatives entirely. They will still be necessary at certain junctures to solve ODEs. Definition The order of an ODE refers to the highest order derivative taken of the dependent variable. Note that notations for higher order derivatives include y (n) and dny 0 dx n , and we will use both. We also use y for the first derivative 00 and y for the second derivative. Example Identify the order of the following ODEs. d 2y dx 2 6y (6) 4 1. ( dy dx ) + = 2. y (4) + + 3. y0 = y 00 −2y y +x d 3y +y dx 3 y 00 = 0 −x (third) (sixth) (second) Definition Let x0 be real number in the domains of y , y 0 , y 00 , . . . , and y (n) (x). Further, let y0 , y1 , y2 , . . . yn−1 be real numbers, as well. An nth order initial value problem is an nth order ODE with n additional bits of information about y , and can be put in the following form: y (n) = f (x, y , y 0 , y 00 , . . . , y (n−1) ), y (x0 ) = y0 , y 0 (x0 ) = y1 , y 00 (x0 ) = y2 , .. . y (n−1) (x0 ) = yn−1 . Example of a First-order IVP dy = y + x, y (0) = 1 dx Example of a Second-order IVP y 00 + y = 0, y (0) = 1, y 0 (0) = 0 Example of a Third-order IVP y 00 + x d 3y = , dx 3 y0 + y y (0) = 1, y 0 (0) = 1, y 00 (0) = −1