Ordinary differential equation (ODE) means that the unknown quantity of differential equation is a function of a single independent variable. In the simplest ordinary differential equation, the unknown quantity is a real function or a complex function, but it may also be a vector function or a matrix function, which can correspond to a system composed of ordinary differential equations. The general expression of differential equation is:
Ordinary differential equations are often classified by order, and order refers to the highest order number of derivatives of independent variables. The two most common types are first-order differential equations and second-order differential equations. For example, the following Bessel equation:
(where y is the dependent variable) is a second-order differential equation whose solution is Bessel function.
Partial differential equation (PDE) means that the unknown quantity of a differential equation is a function of several independent variables, and there is partial differential of the unknown quantity to the independent variables in the equation. The definition of order of partial differential equations is similar to ordinary differential equations, but it is further subdivided into elliptic, hyperbolic and parabolic partial differential equations, especially second-order partial differential equations. Some partial differential equations cannot be classified into any of the above categories in the whole range of independent variables, and such partial differential equations are called mixed types. Equations like the following are partial differential equations:
Linear and nonlinear
Ordinary differential equations and partial differential equations can be divided into linear and nonlinear categories.
If the independent variable and differential term have no square or other product terms, and the dependent variable and differential term have no product, the differential equation is linear, otherwise it is nonlinear.
Homogeneous linear differential equation is a finer classification of linear differential equation, and the result of multiplying the solution of differential equation by a coefficient or adding another solution is still the solution of differential equation.
If the coefficient of a linear differential equation is constant, it is a linear differential equation with constant coefficient. Linear differential equations with constant coefficients can be transformed into algebraic equations by Laplace transform, thus simplifying the solution process.
For nonlinear differential equations, there are few methods to obtain analytical solutions of differential equations, and these methods require special symmetry of differential equations. For a long time, nonlinear differential equations may have very complex characteristics, and there may also be chaos. Some basic problems about nonlinear differential equations, such as the existence and uniqueness of solutions, the well-posedness of initial value problems of nonlinear differential equations, and the boundary value problems of nonlinear differential equations, are quite difficult problems. Even for the above-mentioned basic problems of specific nonlinear differential equations, it is a breakthrough in mathematical theory. For example, among the seven Millennium Prize puzzles put forward in 2000, one is the existence and smoothness of the Navier-Stokes equation, and the mathematical properties of its solutions are discussed. As of August 20 18, this problem has not been proved.
Linear differential equations are often used to approximate nonlinear differential equations, but they can only be approximated under certain conditions. For example, the motion equation of a simple pendulum is a nonlinear differential equation, but it can be approximated as a linear differential equation at a small angle.