Slide 1

• Topic: Differential Equations - Meaning of Mdx + Ndy = 0
• Introduction to differential equations
• Meaning of Mdx + Ndy = 0
• Standard form of a differential equation
• Examples of differential equations
• Importance of differential equations in various fields

Slide 2

• Definition of a differential equation
• Brief explanation of dependent and independent variables
• Order of a differential equation
• Solution of a differential equation
• General solution vs. particular solution

Slide 3

• Different types of differential equations
  • Ordinary differential equations (ODEs)
  • Partial differential equations (PDEs)
• Examples of ODEs and PDEs
• Importance of classifying differential equations

Slide 4

• Linear vs. nonlinear differential equations
• Definition of a linear differential equation
• Example of a linear differential equation
• Definition of a nonlinear differential equation
• Examples of nonlinear differential equations

Slide 5

• Homogeneous vs. non-homogeneous differential equations
• Definition of a homogeneous differential equation
• Example of a homogeneous differential equation
• Definition of a non-homogeneous differential equation
• Example of a non-homogeneous differential equation

Slide 6

• First-order linear differential equations
• General form of a first-order linear differential equation
• Solving first-order linear differential equations using integrating factor method
• Example of a first-order linear differential equation and its solution
• Applications of first-order linear differential equations

Slide 7

• Second-order linear differential equations
• General form of a second-order linear differential equation
• Solving second-order linear homogeneous differential equations using characteristic equation method
• Example of a second-order linear homogeneous differential equation and its solution

Slide 8

• Solving second-order linear non-homogeneous differential equations using particular solution and complementary function
• Example of a second-order linear non-homogeneous differential equation and its solution
• Application of second-order linear differential equations in mechanical systems

Slide 9

• Bernoulli’s differential equation
• General form of Bernoulli’s differential equation
• Solving Bernoulli’s differential equation using substitution method
• Example of a Bernoulli’s differential equation and its solution
• Applications of Bernoulli’s differential equation

Slide 10

• Euler’s differential equation
• General form of Euler’s differential equation
• Solving Euler’s differential equation using substitution method
• Example of an Euler’s differential equation and its solution
• Applications of Euler’s differential equation

Slide 11

• Applications of differential equations in physics
  • Newton’s law of cooling
  • Radioactive decay
  • Harmonic motion
• Examples of differential equations in physics
• Importance of differential equations in understanding physical phenomena

Slide 12

• Applications of differential equations in biology
  • Population dynamics
  • Epidemiology
  • Neural dynamics
• Examples of differential equations in biology
• Importance of differential equations in modeling biological systems

Slide 13

• Applications of differential equations in economics
  • Economic growth models
  • Stock market predictions
  • Interest rate modeling
• Examples of differential equations in economics
• Importance of differential equations in understanding economic trends

Slide 14

• Applications of differential equations in engineering
  • Electrical circuit modeling
  • Fluid dynamics
  • Control systems
• Examples of differential equations in engineering
• Importance of differential equations in designing and optimizing engineering systems

Slide 15

• Applications of differential equations in computer science
  • Image and signal processing
  • Machine learning algorithms
  • Cryptography
• Examples of differential equations in computer science
• Importance of differential equations in developing computational methodologies

Slide 16

• Applications of differential equations in finance
  • Options pricing models
  • Portfolio optimization
  • Risk management
• Examples of differential equations in finance
• Importance of differential equations in quantitative finance

Slide 17

• Techniques for solving first-order differential equations
  • Separable variables
  • Exact equations
  • Integrating factors
• Examples of solving first-order differential equations using different techniques

Slide 18

• Techniques for solving second-order homogeneous differential equations
  • Characteristic equation
  • Method of undetermined coefficients
  • Variation of parameters
• Examples of solving second-order homogeneous differential equations using different techniques

Slide 19

• Techniques for solving second-order non-homogeneous differential equations
  • Method of undetermined coefficients (particular solution)
  • Variation of parameters (complete solution)
• Examples of solving second-order non-homogeneous differential equations using different techniques

Slide 20

• Applications of differential equations in environmental science
  • Population ecology
  • Climate modeling
  • Pollution dynamics
• Examples of differential equations in environmental science
• Importance of differential equations in studying and managing environmental issues

Slide 21

• Techniques for solving higher-order linear differential equations
  • Method of undetermined coefficients
  • Variation of parameters
• Examples of solving higher-order linear differential equations using different techniques

Slide 22

• Applications of differential equations in medicine
  • Pharmacokinetics
  • Neurology
  • Tumor growth modeling
• Examples of differential equations in medicine
• Importance of differential equations in understanding medical processes

Slide 23

• Techniques for solving system of linear differential equations
  • Matrix method
  • Eigenvalue method
  • Laplace transform method
• Examples of solving system of linear differential equations using different techniques

Slide 24

• Applications of differential equations in chemistry
  • Chemical reaction kinetics
  • Thermodynamics
  • Quantum mechanics
• Examples of differential equations in chemistry
• Importance of differential equations in studying chemical systems

Slide 25

• Laplace transform method for solving differential equations
• Definition and properties of Laplace transform
• Solving first-order differential equations using Laplace transform
• Solving second-order differential equations using Laplace transform
• Applications of Laplace transform in electrical circuits

Slide 26

• Fourier series and Fourier transform for solving differential equations
• Definition and properties of Fourier series
• Solving partial differential equations using Fourier series
• Definition and properties of Fourier transform
• Solving differential equations using Fourier transform

Slide 27

• Numerical methods for solving differential equations
  • Euler’s method
  • Runge-Kutta method
  • Finite difference method
• Examples of solving differential equations numerically using different methods

Slide 28

• Stability analysis of differential equations
  • Definition of stability
  • Stability analysis for linear differential equations
  • Stability analysis for nonlinear differential equations
• Examples of stability analysis of differential equations

Slide 29

• Bifurcation theory in differential equations
• Definition and types of bifurcations
• Studying bifurcations using phase portraits
• Examples of bifurcations in differential equations
• Importance of bifurcation theory in understanding dynamic systems

Slide 30

• Summary of key concepts in differential equations
• Importance of differential equations in various fields
• Applications of differential equations in science and engineering
• Techniques for solving differential equations
• Numerical methods and stability analysis of differential equations
• Bifurcation theory and advanced topics in differential equations