Lecture on Differential Equations by Gajendra Purohit

Introduction

• Gajendra Purohit, YouTube channel for engineering mathematics and BSc.
• Useful for competitive exams requiring higher mathematics.
• Announcement of 2.0 version of previously uploaded content.

Basics of Differential Equations

• Definition: An equation involving derivatives of a dependent variable with respect to an independent variable.
• Solution: To solve, integrate the differential equation.
  • 1st derivative: 1 integration → 1 constant.
  • 2nd derivative: 2 integrations → 2 constants.
• Integral Equation: Remove the integration sign using differentiation.
  • If differentiation to be removed, perform integration.

Types of Differential Equations

• Ordinary Differential Equations (ODE): Involves derivatives of one dependent variable with respect to one independent variable.
• Partial Differential Equations (PDE): Involves derivatives of one dependent variable with respect to multiple independent variables.

Understanding Variables

• Dependent Variable: Denoted as the upper variable.
• Independent Variable: Denoted as the lower variable.

Derivatives

• Ordinary Derivatives: When involving one independent variable.
• Partial Derivatives: When involving multiple independent variables.
  • Example: A father with three children (x, y, z), where derivatives are taken with respect to one independent variable treating others as constants.
  • Notation of partial derivatives must be understood.

Order and Degree of Differential Equations

• Order: The highest order derivative present in the differential equation.
  • Example: If the highest derivative is $d^3y/dx^3$, it's a 3rd order differential equation.
• Degree: The power of the highest order derivative after removing fractions or irrational powers.
  • Example: If the highest derivative $d^3y/dx^3$ has power 2, the degree is 2.
  • Non-existent Degree: If terms in a differential equation make it impossible to determine degree (fractions, irrational powers).

Linear and Non-linear Differential Equations

• Linear Differential Equation: Derivative variables' power is 1 and no product of dependent variables and their derivatives.
  • Example: (1st degree in derivative).
• Non-linear Differential Equation: Derivative variables' power is greater than 1 or products of dependent variables and their derivatives.
  • Example: If power of derivative is 2 or more.

Examples and Practice

• Example 1: Identify order and degree.
  • Highest order derivative: 1 → Order 1,
  • Degree: Highest power, usually after removing fractions.
• Example 2: Order 2, with fraction powers to be removed for finding degree.
  • Highest order derivative: 2 → Order 2,
  • Degree: Power of 3 after removing fraction.

Closing Remarks

• Playlist and new content available for better understanding and preparation.
• Follow and subscribe for more educational content.
• Interactive question for viewers: Identify the order and degree of a given differential equation.

Thank you for watching, like, share, and subscribe!