Differential Equations-I (Grade A+)

Summary:

Differential equations are mathematical equations that involve functions and their derivatives. They can be categorized as either ordinary differential equations (ODEs) or partial differential equations (PDEs), depending on whether they involve only ordinary derivatives or partial derivatives, respectively. The order of a differential equation corresponds to the highest-order derivative present in the equation.

In the study of differential equations, it is essential to understand the concept of solutions. A solution to a differential equation is a function that satisfies the equation for every point in a given domain. Solutions can be classified as particular solutions or general solutions. A particular solution satisfies the equation for specific initial or boundary conditions, while a general solution encompasses all possible solutions.

Differential equations play a crucial role in various fields, and their applications are diverse. They can be used to model real-world phenomena and solve problems in physics, engineering, biology, economics, and many other disciplines. Solving differential equations involves formulating a mathematical model based on a real-world situation, finding the solution to the model, and interpreting the results in the context of the problem.

Different techniques for solving differential equations, depending on their type and characteristics. Some common methods include separating variables, integrating factors, power series solutions, and numerical methods. Each technique has advantages and is suitable for different equations and situations.

Excerpt:

Differential Equations-I

Chapter 1

Introduction

1.1 Preliminaries
Definition (Differential equation)
A differential equation (de) involves a function and its derivatives.

Differential equations are called partial differential equations (PDE) or ordinary differential equations (ode) according to whether or not they contain partial derivatives. The order of a differential equation is the highest order derivative occurring. A solution (or particular solution) of a differential equation of order n consists of a function defined and n times differentiable on a domain D having the property that the functional equation obtained by substituting the function and its n derivatives into the differential equation holds for every point in D.