Topic outline

Ordinary Differential Equation

Introduction:

In this class, an Ordinary differential equation will be discussed. In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.

Objectives:

After studying this unit you will be able to

• know about the formation of differential equation
• use equations to solve a real-life problem

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

Order, degree, and variable separable

Introduction:

In this class, the order and degree of a differential equation will be discussed.

Let a differential equation be expressed as a polynomial equation in the derivatives occurring in it.

The exponent of each of the derivatives should be minimum. The exponent of the variables need not be an integer.
Under such conditions,
i) the order of the differential equation is the order of the highest derivative in it.
ii) the degree of the differential equation is the largest exponent of the highest order derivative

Separable equations introduction. "Separation of variables" allows us to rewrite differential equations so we obtain equality between two integrals we can evaluate. Separable equations are the class of differential equations that can be solved using this method.

Objectives:

• order and degree of ordinary differential equation
• solve 1st order ODE by using variable separation method

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

Homogeneous Differential Equation

Introduction:

In this class, a Homogeneous differential equation will be discussed.

A differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term. A homogeneous differential equation is of prime importance in the physical applications of mathematics due to its simple structure and useful solution.

Objectives:

• solving method of homogeneous equations
• application of homogeneous equation

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

First-order differential equation and integrating factor

Introduction:

In this class, the first order and integrating factor of differential equations will be discussed.

In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where the temperature becomes the integrating factor that makes entropy an exact differential.

Objectives:

• to generalize the integrating factor method from linear scalar differential equations to a linear system of differential equations
• to solve first-order linear differential equation
• to make an exact differential equation
• solving the real-life problem by IF method

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

Bernoulli differential equation:

Introduction:

In this class, the Bernoulli differential equation will be discussed.

In mathematics, an ordinary differential equation of the form   is called a Bernoulli differential equation where  is any real number other than 0 or 1. It is named after Jacob Bernoulli, who discussed it in 1695. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A famous special case of the Bernoulli equation is the logistic differential equation.

Objectives:

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

Linear differential equation with constant coefficients

Introduction:

In this class, a linear differential equation with constant coefficients will be discussed.
If the constant term is the zero function, then the differential equation is said to be homogeneous, as it is a homogeneous polynomial in the unknown function and its derivatives. ... A differential equation has constant coefficients if only constant functions appear as coefficients in the associated homogeneous equation.

Homogeneous equation with constant coefficients
A homogeneous linear differential equation has constant coefficients if it has the form

where   are (real or complex) numbers. In other words, it has constant coefficients if it is defined by a linear operator with constant coefficients.

Non-homogeneous equation with constant coefficients:
A non-homogeneous equation of order n with constant coefficients may be written

where  are real or complex numbers, f is a given function of x, and y is the unknown function (for sake of simplicity, "(x)" will be omitted in the following). There are several methods for solving such an equation. The best method depends on the nature of the function f that makes the equation non-homogeneous. If f is a linear combination of exponential and sinusoidal functions, then the exponential response formula may be used.

Objectives:

• Solve the homogeneous linear differential equations with constant coefficients.
• Apply the method to solve the non-homogeneous linear differential equations with constant coefficients.
• solve systems of linear differential equations.
• determine the type of linear differential equation systems
• Use the operator method to solve linear systems with constant coefficients.
• Solve the linear systems in normal form.
• Solve the homogeneous linear systems with constant coefficients.
Lesson Plan:
• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

Initial value problem

Introduction:

In this class, the Initial value problem will be discussed.

In the field of differential equations, an initial value problem (also called a Cauchy problem by some authors) is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution. In physics or other sciences, modeling a system frequently amounts to solving an initial value problem; in this context, the differential initial value is an equation that is an evolution equation specifying how, given initial conditions, the system will evolve with time.

Objectives:

• Recognize an initial value problem
• solve initial value problem of the linear differential equation with constant coefficients
• application of initial value problem

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

Application of ODE

Introduction:

In this class, the application of ODE will be discussed.

Objectives:

• to know about real-life application of ODE: Exponential Growth - Population, Exponential Decay - Radioactive Material, Falling Object, Newton's Law of Cooling, RL circuit

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

Introduction to Partial differential equation

Introduction:

In this class, the Partial differential equation will be discussed.

A partial differential equation (or briefly a PDE) is a mathematical equation that involves two or more independent variables, an unknown function (dependent on those variables), and partial derivatives of the unknown function with respect to the independent variables. The order of a partial differential equation is the order of the highest derivative involved. A solution (or a particular solution) to a partial differential equation is a function that solves the equation or, in other words, turns it into an identity when substituted into the equation. A solution is called general if it contains all particular solutions of the equation concerned.

Partial differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc.

Objectives:

• equip with the concepts of partial differential equations and how to solve linear Partial Differential with different methods.
• introduce to some physical problems in Engineering models that result in partial differential equations

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

Formation of PDE:

Introduction:

In this class, the formation of PDE will be discussed.

Partial differential equations can be obtained by the elimination of arbitrary constants or by the elimination of arbitrary functions.

Objectives:

• form partial differential equation by eliminating constant
• understand the properties partial differential equation

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

The linear partial differential equation of order one

Introduction:

In this class, the Linear partial differential equation of order one will be discussed. In mathematics, a first-order partial differential equation is a partial differential equation that involves only the first derivatives of the unknown function of n variables. The equation takes the form

Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems, and in simple models for gas dynamics whose solution involves the method of characteristics. If a family of solutions of a single first-order partial differential equation can be found, then additional solutions may be obtained by forming envelopes of solutions in that family. In a related procedure, general solutions may be obtained by integrating families of ordinary differential equations.

Objectives:

• to know about the method of characteristics
• solve by using the Lagrange method

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

The linear partial differential equation of order two

Introduction:

In this class, the Linear partial differential equation of order two will be discussed. A second-order linear partial differential equation with two independent variables has the form

Objectives:

•  solve a different kind of problem
• second-order partial differential equations can be classified as parabolic, hyperbolic, and elliptic.

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

Introduction:

In this class, the Partial differential equation with constant coefficients will be discussed.

Objectives:

•  to solve higher order partial differential equations

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work

Application of PDE

Introduction:

In this class, the application of PDE will be discussed. Partial differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc.

Objectives:

• to know about real-life application of PDE

Lesson Plan:

• Discussion from a lecture slide
• Problem-solving
• Group Work
• Home Work