Because most differential equations can't get explicit solutions, we just analyze the stability of their solutions or get approximate numerical solutions. This part is very rich and there is a lot of work to be done.
If the unknown function in a differential equation is a multivariate function and there is a partial derivative operation of the unknown function, the equation is called a partial differential equation.
After the establishment of calculus, the related theories of ordinary differential equations developed rapidly. Ordinary differential equations are also applied to the discussion of geometric and mechanical problems, and explain the facts in celestial mechanics that have been known in the early days, and obtain new discoveries.
The study of algebra has entered the stage of overall analysis of system structure from the study of locality. Since the emergence of group theory, there have been various branches of mathematics, such as lattice theory of ordered structure, topology, modular theory of composite structure of rings and groups, topological vector space, differential manifolds and fiber bundles with several structures at the same time. It can be said that structural thought is one of the most basic and important ideas in all branches of modern mathematics.
Partial differential equations are closely related to other branches of mathematics, such as functional analysis, function theory, topology, algebra and complex analysis, and the basic concepts, ideas and methods in these branches of mathematics have been widely used.
Usually, the equation with unknown function under the integral symbol is called an integral equation, and if the unknown function is a multivariate function, the equation is called a multidimensional integral equation.