Differential equation refers to the relationship with unknown function and its derivative. Solving differential equations means finding unknown functions.
Differential equations are developed by calculus. Calculus books have dealt with problems related to differential equations. Differential equations are widely used and can solve many problems related to derivatives. Many kinematics and dynamics problems involving variable forces in physics, such as falling bodies with air resistance as speed function, can be solved by differential equations. In addition, differential equations have applications in chemistry, physics, engineering, astronomy and other fields.
Elementary algebra further develops in two directions: linear equations with many unknowns; Higher order equations with higher unknowns. The development of these two directions makes algebra develop to the stage of advanced algebra. Advanced algebra, as a general term for the development of algebra to an advanced stage, includes many branches. Higher algebra offered by universities now generally includes two parts: linear algebra and polynomial algebra.
Using power series as a tool, the function theory is expanded through strict pure analytical reasoning. Analytic function is defined as a function that can be expanded into power series, and the properties of function are studied around singularity.
Functional analysis is a branch of studying the mapping from topological linear space to topological linear space satisfying various topological and algebraic conditions. It is developed from the study of variational problems, integral equations and theoretical physics. It comprehensively uses the viewpoints of function theory, geometry and modern mathematics to study functions, operators and limit theory in infinite dimensional vector space. It can be regarded as analytic geometry and mathematical analysis of infinite dimensional vector space.