4 1. Differential Equations 1.1 Classification of Differential Equations One important distinction to make in a differential equation is whether all of the derivatives are taken with respect to the same variable. A differential equation is called an ordinary differential equation (ODE) if there is only one variable and called a partial differential equation (PDE) if differentiation occurs with respect to more than one variable. For instance, both of these are ordinary differential equations: f(x) = 2f (x) d2y dt2 + y = dy dt 2 while these are partial differential equations: ∂f ∂x + ∂f ∂y = x2 + f uy = uux. Note that a solution to this last equation would be a function u(x, y) such that its derivative with respect to y is equal to the product of the function with its own x-derivative for all values of the variables x and y. A more important distinction is between linear and nonlinear equations. An equation is linear if the unknown function3 and its derivatives are multiplied by coefficients and added together, but never multiplied by each other. Note that of the four differential equations listed above in this section, the first one in each pair is linear. Also, the second equation in each pair is nonlinear. (One is nonlinear because the derivative dy/dt is squared and in the other case it is because u is multiplied by ux.) We will see more about the difference between linear and nonlinear equations in Chapter 2. Finally, another important way to classify differential equations is to identify whether the coefficients in the equation are constants or depend on the variables. For instance, this is a big difference between equations (1.5) and (1.6). In the first one the coefficients (1 + x) and x depend on the variable x while in the second equation all of the co- efficients are constant. We say that the equation is autonomous if the coefficients are constant (and that it is nonautonomous otherwise). 3 It is often assumed in this book that a differential equation involves only one undetermined function and that the rest are known coefficients. In fact, there are also differential equations that involve more than one function and so are solved by collections of functions, but we will not generally consider such equations.
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