BSc (NEP) - IV Sem - Matrices & Differential Equations

Differential EQUATIONS

Semester 4th

A Comprehensive Guide

B.Sc. Mathematics - IV, Part A

Defferential Equation

Differential Equations

A differential equation is a mathematical equation that relates a function with its derivatives. It describes how a quantity changes over time or space, making it a powerful tool in modeling real-world phenomena. Differential equations appear in physics, engineering, economics, biology, and many other fields.

1. Definition of a Differential Equation

• A differential equation (DE) is an equation that contains one or more derivatives of an unknown function. It can be written in the general form:

  $$F(x, y, y', y'', ..., y^{(n)}) = 0$$

  where:

• $x$ is the independent variable (e.g., time, distance, etc.).
• $y$ (or $f(x)$) is the dependent variable (the function we want to find).
• $y' = \frac{dy}{dx}$, $y'' = \frac{d^2y}{dx^2}$ etc., represent the derivatives of $y$.

2. Types of Differential Equation

A. Ordinary vs. Partial Differential Equations

1. Ordinary Differential Equations (ODEs)

   • Involve derivatives of a function with respect to only one independent variable.
   • Example: $$\frac{dy}{dx} + 3y = 5$$Here, $x$ is the independent variable, and $y$ is the dependent variable.

2. Partial Differential Equations (PDEs)

• Involve derivatives with respect to multiple independent variables.
• Example: $$\frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2}$$ This equation, called the heat equation, models heat flow over time.

B. Order of a Differential Equation

• The order of a DE is determined by the highest derivative present in the equation.
• Example:
  • $\frac{dy}{dx} + 2y = 0$ → First-order DE
  • $\frac{d^2y}{dx^2} + 3 \frac{dy}{dx} + 2y = 0$ → Second-order DE

C. Linear vs. Nonlinear Differential Equations

• Linear DE: The dependent variable and its derivatives appear only in the first power and are not multiplied together.
  • Example: $y'' + 5y' + 6y = 0$ (linear because each term is first-degree in $y$).
• Nonlinear DE: Contains products or powers of the dependent variable or its derivatives.
• Example: $y'' + y^2 = 0$ (nonlinear due to $y^2$).

3. Solving Differential Equations

A. General and Particular Solutions

• A general solution contains arbitrary constants and represents a family of solutions.
• A particular solution is obtained by applying initial or boundary conditions.

B. Methods of Solving DEs

1. Separation of Variables (for first-order DEs)

   • Example: Solve $\frac{dy}{dx} = ky$.
   • Separate: $\frac{dy}{y} = k dx$.
   • Integrate: $\ln y = kx + C$.
   • Solution: $y = Ce^{kx}$.

2. Integrating Factor Method (for linear first-order DEs)

   • Standard form: $\frac{dy}{dx} + P(x)y = Q(x)$.
   • Multiply by an integrating factor $e^{\int P(x)dx}$to simplify.

3. Characteristic Equation (for second-order linear DEs)

   • Example: Solve $y'' - 3y' + 2y = 0$.
   • Solve characteristic equation $r^2 - 3r + 2 = 0$.
   • Find roots and construct solution.

4. Laplace Transform (for more complex problems, especially in engineering).

4. Applications of Differential Equations

Physics & Engineering

• Newton’s Second Law: $F = ma$ leads to a second-order DE in motion analysis.
• Electrical Circuits: $L\frac{dI}{dt} + RI = V$ (models current in an inductor).
• Heat Equation, Wave Equation, Fluid Flow models.

Biology & Medicine

• Population Growth: $\frac{dP}{dt} = rP$ (exponential growth).
• Spread of Diseases: $\frac{dI}{dt} = \beta SI - \gamma I$ (SIR model).

Economics & Finance

• Interest rate models: Black-Scholes equation for option pricing.
• Logistic Growth in markets: $\frac{dP}{dt} = rP(1 - \frac{P}{K})$
  

Real-Life Applications of Matrices

Matrices play a crucial role in various real-world scenarios, including:

• Google PageRank Algorithm: Google uses matrices to rank web pages based on relevance and connectivity.
• 3D Graphics & Animations: Used in gaming and movies to perform transformations such as scaling and rotation.
• Cryptography: Matrices help encode and decode secure messages in cybersecurity.
• Weather Prediction: Meteorologists use matrices to model climate patterns and make forecasts.
• Traffic Control Systems: Matrices analyze and optimize traffic flow in urban areas.

  Syllabus

(NEP SYLLABUS MODIFIED BY CCS UNIVERSITY)

Differential Equations

B.Sc. MATHEMATICS SEMESTER-II PART A

AS PER NEP 2020 [NATIONAL EDUCATION POLICY]

B.A./B.Sc. Paper-I : Part A

Unit I

Second order linear differential equations with variable coefficients: The complete Solution in terms of A known Integral, Removal of the first order Derivative (normal form), Solution by Changing the Independent Variable, variation of parameters, Method of Operational Factors.

Unit II

Bessel and Legendre functions and their properties, Orthogonal properties, recurrence Formula and generating Function.

Unit III

Origin of first order partial differential equations. Partial differential equations of the first order and degree one, Lagrange's solution, Partial differential equation of first order and degree greater than one. Charpit's method of solution, Surfaces Orthogonal to the given system of surfaces.

Unit IV

Origin of second order PDE, Solution of partial differential equations of the second and higher order with constant coefficients, Classification of linear partial differential equations of second order, Solution of second order partial differential equations with variable coefficients, Monge's method of solution.

Book Chapters Name

Chapter 1- Linear Equations of Second Order with Variable Coefficients (Solution)

Chapter 2- Partial Differential Equations of the First Order (Coming Soon)

Chapter 3- Linear Partial Differential Equations of second Higher Order With Constant Coefficients (Coming Soon)

Chapter 4- Partial Differential Equations of Second Order with Variable Coefficients (Coming Soon)

Chapter 5- Monge's Method(Coming Soon)

Chapter 6- Legendre's Functions(Coming Soon)

Chapter 7- Bessel's Functions(Coming Soon)

Chapter 8- Hypergeometric Functions(Coming Soon)

Chapter 9- Series Solutions of Differential Equations(Coming Soon)