Differential Equations 2.0

Introduction

• Presenter: Gajendra Purohit
• Content: Engineering Mathematics (Specifically Differential Equations)
• Target Audience: Students preparing for competitive exams, engineering, and BSc students
• Announcement: Upgrading old content (started in 2018) to version 2.0

Differential Equations: Basic Terms

• Definition: An equation involving derivatives is called a differential equation.
• Solution: To solve a differential equation, integrate the equation.
  • Single derivative: one-time integration
  • Double derivative: two-time integration
• Integral Equation: Remove integration sign using derivatives.
• Differential Equation: Remove differentiation using integration.

Understanding Variables

• Dependent Variable: Upper one (typically represented as a function of the independent variable)
• Independent Variable: Lower one (the variable that the function is derived with respect to)

Types of Differential Equations

Ordinary Differential Equation (ODE)

• Definition: Dependent variable with single independent variable
• Example: Derivatives involve only one independent variable

Partial Differential Equation (PDE)

• Definition: Dependent variable with multiple independent variables (e.g., x, y, z)
• Analogy: Father with multiple children (each child representing different independent variables)
• Example: Involves partial derivatives with respect to different independent variables

Order and Degree of Differential Equations

Order

• Definition: Highest order derivative present in the differential equation
• Example:
  • Equation with highest derivative 3 → 3rd order
  • Equation with highest derivative 1 → 1st order

Degree

• Definition: Power of the highest order derivative, after removing any fractional powers
• Example:
  • Equation with highest order 3 and power 1 → Degree is 1
  • Equation with highest order 3 and power 2 → Degree is 2
• Important Concept: In some cases, degree does not exist (e.g., fractional powers in the highest derivative)

Linear vs Non-Linear Differential Equations

Linear Differential Equations

• Definition: Derivative's power is 1 and variable functions are linear
• Example: Linear term's power is 1

Non-Linear Differential Equations

• Definition: Derivative's power is greater than 1 or involves non-linear terms
• Example: Derivative’s power is 2
  • Non-linear partial differential equation

Practice Questions

• Question Example: Find the order and degree of a given differential equation.
• Key Steps:
  • Identify the highest order derivative
  • Determine the degree by removing fractional powers

Conclusion

• Upgrading playlist for better understanding
• Viewer Interaction: Questions in the comment box
• Additional Resources: Links to playlists, new channel for short tricks, Instagram
• Call to Action: Like, share, and subscribe