First, the method of separating variables.
Variable separation method of differential equation is a method to solve partial differential equation. It can decompose a partial differential equation into two or more ordinary differential equations with only one variable, thus splitting the original equation into several simpler ordinary differential equations with only one independent variable. Solve these ordinary differential equations one by one, combine the solutions, and finally get the solution of the original partial differential equation.
The method of separating variables of differential equations is widely used in the field of mathematical physics, which can solve many practical problems, such as heat conduction equation, wave equation, Laplace equation and so on. By expressing the unknown function as the product of variables, the partial differential equation is transformed into an ordinary differential equation, which makes the original difficult problem simple and feasible.
Second, the constant variation method
The constant variation method of differential equation is an effective method to solve linear differential equation. It changes the constant c in the solution of homogeneous equation into a function u(x) about x, and then replaces u(x) with the original equation to get a differential equation about u(x), thus transforming the original equation into a first-order linear differential equation to solve 1.
Specifically, for the first-order non-homogeneous linear differential equation in the form of "y'+p(x)y=q(x)", we can assume "y = ce {-\ int p (x) dx}", change c into a function u(x) about x, and get "y = u (x) e.
Constant variation method is a special variable substitution method, which can transform high-order differential equations into low-order differential equations and make the solution process more concise and feasible.
Third, variable substitution method.
1. Introduce a new unknown function or independent variable, and use the new unknown function or independent variable to represent the unknown function or independent variable in the original equation, thus transforming the original equation into a more manageable equation.
2. Introduce a new independent variable, and use the new independent variable to represent the independent variable in the original equation, thus transforming the original equation into a more manageable equation. In practical application, different differential equations can be selected and combined according to their types and solving needs, thus simplifying calculation and improving solving efficiency.
On the Function of Differential Equation
1, the role of differential equations in science and engineering
In physics, differential equations are used to describe the motion of objects, the change of electromagnetic fields, heat conduction and other phenomena. In chemistry, differential equations are used to describe the dynamic process of chemical reactions. In the engineering field, differential equations are used to describe the behavior of various systems, such as mechanical systems, electrical systems, control systems and so on.
By solving differential equations, we can predict the behavior of the system and optimize the performance of the system, thus promoting the development of science and technology.
2. The role of differential equations in the economic field.
In economics, differential equations are used to describe the changing trend of economic variables, such as population growth, price change and economic growth. By establishing a differential equation model, we can predict the development trend of economic phenomena and provide a basis for formulating policies.
In addition, differential equations are also widely used in the financial field, such as changes in stock prices and optimization of investment strategies. By solving the differential equation, we can predict the market trend and provide support for investment decision.