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With the introduction of differential operation to the function in the analysis, the equation representing the relationship between the derivative and the independent variable of the unknown function has entered the mathematician's field of vision, which is the differential equation. The formation and development of differential equations are closely related to the development of science and technology such as mechanics, astronomy and physics. Because in the real world, the movement of matter and its changing law are mathematically described by functional relations, that is to say, the solution of a problem is to seek a function that meets certain conditions, and such a problem is transformed into the solution of a differential equation. Differential equations provide powerful tools for scientific discovery, such as:
Newton used differential equations to study celestial mechanics and mechanical mechanics, and theoretically obtained the laws of planetary motion;
Astronomers Adam and Le Verrier discovered Neptune by using differential equations.
The basic idea of solving differential problems is similar to solving algebraic equations. It is necessary to find out the relationship between the known function and the unknown function in the problem, and then get one or more equations containing the unknown function, and then get the expression of the unknown function by analytical method.
If an unknown function in a differential equation contains only one independent variable, then this kind of differential equation is an ordinary differential equation. The general solution of ordinary differential equations constitutes a function family, which mainly studies the classification and solution of equations or equations, the existence and uniqueness of solutions, singular solutions, qualitative theory and so on.
Functional for solving ordinary differential equations
Now, ordinary differential equations have important applications in automatic control, the design of various electronic devices, the calculation of trajectory, the study of flight stability of aircraft and missiles, and the study of the stability of chemical reaction process.