Lecture on Differential Equations by Dr. Kanjin Purohit

Introduction

• Dr. Kanjin Purohit's YouTube channel for Engineering Mathematics, B.Sc., and Competitive Exam Preparation.

Basic Terms in Differential Equations

• Differential equation involving derivatives.
• Example: For a circle equation x^2 + y^2 = a^2.
  • Differentiate w.r.t. x: Results in a differential equation.
• Solving a differential equation means integrating it back to the original function.
  • Single Derivative: Integrate once.
  • Double Derivative: Integrate twice.

Dependent and Independent Variables

• Dependent Variable: The variable whose changes depend on another variable (e.g., y in dy/dx).
• Independent Variable: The variable on which the dependent variable depends, usually x.
• Ordinary Differential Equations (ODEs): Involve a single independent variable.
• Partial Differential Equations (PDEs): Involve multiple independent variables.

Ordinary vs. Partial Differential Derivatives

• Ordinary Derivatives: Derivatives w.r.t. a single independent variable (dy/dx).
• Partial Derivatives: Derivatives w.r.t. one variable while keeping others constant.
  • Example: In a system with x, y, and z, partial derivatives such as ∂/∂x while keeping y and z constant.

Order and Degree of Differential Equations

• Order: The highest order derivative in the equation.
  • Example: d^3y/dx^3 (Order = 3).
• Degree: The exponent of the highest-order derivative after making it free from radicals and fractions.
  • Example: d^3y/dx^3 + dy/dx^2 has Degree = 1.
  • Equations involving terms like e^(dy/dx), sin(dy/dx), etc., have no defined degree.

Linear vs. Non-linear Differential Equations

• Linear: Derivatives are to the power of 1 and the function involves no products of the dependent variable's derivatives.
  • Example: dy/dx + sin(x) = 0 (Linear).
  • Examples of Non-linear: dy/dx * y = 0 (Product term), (dy/dx)^2 + sin(x) = 0 (Square of a derivative).*

Examples Discussed

• Example 1: dy/dx + f(x) = 0 with f(x) not involving y directly - Linear.
• Example 2: (d^3y/dx^3)^2 + x^2*(d^2y/dx^2) = 0 - Non-linear.*

Practical Application

• Students are encouraged to solve example problems and share the time taken to solve in the comments section.

Conclusion

• Importance of understanding differential equations for engineering and other scientific fields.
• Encouragement to engage with content, like, share, and subscribe to the channel.