Differential Equations - The Smart Notes
Introduction to Differential Equations
**Differential Equations** are mathematical equations that relate a function with its derivatives. They are essential for modeling real-world phenomena such as population growth, motion of objects, electrical circuits, heat conduction, and fluid dynamics. Differential equations are classified into **ordinary differential equations (ODEs)** and **partial differential equations (PDEs)** based on the number of variables involved.
Types of Differential Equations
- First-Order Differential Equations: Involves the first derivative of the function.
- Second-Order & Higher-Order Equations: Contains second or higher derivatives.
- Linear vs. Nonlinear Equations: Linear equations have terms that are only first-degree, while nonlinear equations involve powers or products of the function and its derivatives.
- Partial Differential Equations (PDEs): Involves functions of multiple variables and their partial derivatives.
- Homogeneous & Non-Homogeneous Equations: Homogeneous equations have zero on one side, while non-homogeneous equations contain non-zero terms.
Applications of Differential Equations
Differential equations play a crucial role in multiple fields:
- Physics: Newton’s Laws of Motion, Thermodynamics, and Electromagnetism.
- Engineering: Electrical circuits, Control Systems, and Mechanical Vibrations.
- Economics: Growth Models, Investment Analysis, and Optimization.
- Biology: Population Growth Models and Epidemic Spread.
- Computer Science: Machine Learning Models and Artificial Intelligence Algorithms.