Introduction to Differential Equations 2.0

Overview of Content

• The speaker introduces a new series on differential equations, building upon previous content from 2018. This updated version aims to provide improved insights for students preparing for engineering or B.Sc. exams.

Key Concepts Discussed

• The session will cover the concepts of order and degree in differential equations, as well as conditions under which the degree may not exist. Students are encouraged to stay engaged for valuable content.

Understanding Differential Equations

Definition and Examples

• A differential equation involves derivatives such as dy/dx or d^2y/dx^2 . An example provided is the equation of a circle: x^2 + y^2 = a^2 . Differentiating this with respect to x leads to a non-differential equation form.

Integration Process

• To solve a differential equation, integration is required when derivatives are involved; one integration introduces a constant into the solution process. If double derivatives are present, two integrations will be necessary. Understanding this concept is crucial for solving differential equations effectively.

Dependent and Independent Variables

Variable Relationships

• In any given differential equation, the dependent variable (e.g., y ) relies on an independent variable (e.g., x ). The relationship can be visualized through examples like pillars where one supports another, illustrating dependency clearly. This understanding helps clarify confusion regarding which variables are dependent or independent in various contexts.

Types of Differential Equations

Classification by Variables

• A key distinction made is between ordinary and partial differential equations based on the number of independent variables involved:

• Ordinary Differential Equation: Involves one independent variable.

• Partial Differential Equation: Involves multiple independent variables but only one dependent variable.

This classification aids in identifying how to approach different types of problems within calculus and physics contexts.

Derivatives Explained

Ordinary vs Partial Derivatives

• When dealing with derivatives:

• An ordinary derivative arises when there’s only one dependent variable.

• A partial derivative occurs when multiple independent variables influence the outcome.

Understanding Differential Equations

Introduction to Derivatives

• The discussion begins with the concept of derivatives, specifically focusing on partial derivatives as a means to understand differential equations.

• An ordinary differential equation (ODE) is defined as one involving derivatives with respect to a single independent variable, while a partial differential equation (PDE) involves multiple independent variables.

Order and Degree of Differential Equations

• The order of a differential equation is determined by the highest order derivative present. For example, if the highest derivative is three, it is classified as a third-order differential equation.

• Understanding how to identify the order of an equation is crucial; it involves recognizing which term contains the highest derivative.

Determining Degree

• The degree of a differential equation refers to the power of its highest order derivative. It’s important to simplify any fractional powers before determining this degree.

• An example illustrates that if an equation has a third-order derivative with power one, its degree will also be one after simplification.

Conditions for Non-existence of Degree

• Certain types of terms in differential equations can lead to non-existent degrees. Specifically, terms like logarithmic or sine functions involving derivatives complicate this determination.

• If an equation includes terms such as log(d/ds) , then it cannot have a defined degree due to their complex nature.

Linear vs Non-linear Differential Equations

• A linear differential equation maintains that all derivatives are in first-degree form without products between dependent variables and their derivatives.

• Examples clarify that if any product exists between dependent variables and their derivatives or if higher powers are involved, the equation becomes non-linear.

Understanding Non-Linear Differential Equations

Introduction to Differential Equations

• The discussion begins with an introduction to differential equations, specifically differentiating between linear and non-linear types.

• A non-linear partial differential equation is presented as an example, emphasizing the confusion students often face regarding linear versus non-linear concepts.

Order and Degree of Differential Equations

• The speaker explains how to determine the order of a differential equation by identifying the highest derivative present, which in this case is one.

• To find the degree, the speaker discusses removing fractional powers from derivatives, leading to a conclusion that the degree is two when squared on both sides.

Example Problem Analysis

• An additional example is provided where students are asked about the order and degree of a given differential equation. The order is determined to be two while explaining how to simplify it for clarity.

• The process involves eliminating fractional powers through manipulation of both sides of the equation, ultimately confirming that the highest order derivative remains two with a degree of three.

Conclusion and Engagement

• The speaker encourages viewers to engage by commenting on their understanding and time taken to solve related questions.