# theory of differential equations

> area of mathematics that deals with theory of differential equations and solutions to them

**Wikidata**: [Q28575007](https://www.wikidata.org/wiki/Q28575007)  
**Source**: https://4ort.xyz/entity/theory-of-differential-equations

## Summary
The theory of differential equations is a branch of mathematics that studies differential equations and their solutions. It is considered both a part of mathematical analysis and a subclass of computational mathematics. The field provides the theoretical framework for understanding equations that describe relationships between a function and its derivatives.

## Key Facts
- **Primary Subject:** The field is the study of the differential equation.
- **Classification:** It is an instance of a branch of mathematics.
- **Parent Fields:** It is considered part of mathematical analysis and a subclass of computational mathematics.
- **ANZSRC 2008 Code:** The field is associated with the Australian and New Zealand Standard Research Classification code `010302` for "Numerical Solution of Differential and Integral Equations".
- **ANZSRC 2020 Code:** The field is associated with the Australian and New Zealand Standard Research Classification code `490303` for "Numerical solution of differential and integral equations".
- **Associated Professionals:** Practitioners in this field are often mathematicians, computer scientists, and university teachers.

## FAQs
### Q: What is the theory of differential equations?
A: It is an area of mathematics focused on the theory behind differential equations and the methods for finding their solutions. It is a foundational branch of both pure and applied mathematics.

### Q: What fields of mathematics is this theory a part of?
A: The theory of differential equations is classified as part of mathematical analysis and as a subclass of computational mathematics.

### Q: What does the theory of differential equations study?
A: Its primary subject of study is the differential equation, which is an equation that relates a function with its derivatives. The theory explores the properties of these equations and their solutions.

## Why It Matters
The theory of differential equations is significant because it provides the fundamental principles for modeling and solving a vast range of problems in science, engineering, and economics. Many real-world phenomena, from the motion of planets to the flow of electricity and the growth of populations, are described by differential equations.

This mathematical theory is crucial for developing methods to find solutions to these equations. Its classification under computational mathematics and its specific research codes for "Numerical Solution of Differential and Integral Equations" highlight its importance in practical applications. By providing a rigorous theoretical foundation, it enables the creation of algorithms and computational tools that can approximate solutions when exact ones are impossible to find, making it an indispensable tool for modern scientific research and technological development.

## Notable For
- **Dual Classification:** It is distinctly classified as both a part of mathematical analysis (a field of pure mathematics) and a subclass of computational mathematics (an applied field).
- **Focus on Theory:** As its name implies, the field is concerned not just with finding solutions, but with the underlying theoretical properties of differential equations themselves.
- **Numerical Application:** The field is specifically recognized in academic research classifications for its role in the numerical solution of differential and integral equations, underscoring its practical importance in computation.

## Body
### Classification in Mathematics
The theory of differential equations is a formal branch of mathematics. It holds a dual position within the broader mathematical landscape:
*   **Part of Mathematical Analysis:** It is considered a component of mathematical analysis, which deals with the concepts of limits, continuity, and convergence.
*   **Subclass of Computational Mathematics:** It is also categorized as a subclass of computational mathematics, which involves the application of mathematical methods to computation.

### Subject and Scope
The central object of study in this field is the **differential equation**. The theory encompasses the formal study of these equations and the various methods used to arrive at their solutions.

### Academic and Research Identifiers
The field is formally recognized in research classification systems, particularly for its applied aspects:
*   **ANZSRC 2008:** Field of Research ID `010302`, named "Numerical Solution of Differential and Integral Equations".
*   **ANZSRC 2020:** Field of Research ID `490303`, named "Numerical solution of differential and integral equations".
*   **Encyclopedia of Modern Ukraine:** It is assigned the identifier `24416`.

### Associated Researchers
Numerous mathematicians and computer scientists have worked within or contributed to the theory of differential equations. Examples include:
*   Dennis Brooke Gannon
*   Germund Dahlquist
*   Evgeny Moiseev
*   Sergey Alekseevich Gusarenko
*   Vladimir Ilyin
*   Vladimir Tretyakov
*   Snehashish Chakraverty

## Schema Markup
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