KNF 1023

ENGINEERING MATHEMATICS II

Synopsis

• The course introduces fundamental knowledge in mathematics that is applicable in the engineering aspect. Topics include First ODE and its engineering application, Second ODE and its application, Laplace Transformation and Fourier Series.

Course Outcomes

Level of Domain PO Assessed

C P A S

CO1 Apply the respective 1st and 2nd order ODE (C3) 3 1

CO2 Analyse engineering problems (growth, decay, flow, spring and series/parallel electronic circuits) using 1st and 2nd order ODE(CT) CT 2

CO3 Utilise the Laplace Transform methods to solve 1st and 2nd order ODEs (CT) CT 1

CO4 Comprehend Fourier Series expansion (C2). 2 1

C: Cognitive ; P: Psychomotor ; A: Affective ; S: Soft-skills (CT: Critical Thinking, TS: Teamwork)

Syllabus

Syllabus

Syllabus

Assessment

Resources

1. D. G. Zill (2009). A First Course in Differential Equations with Modeling Application. 9th Edition. Cengage Learning.

2. G. James (2007). Modern Engineering Mathematics. 4th Edition. Prentice Hall.

3. K. A. Stroud (2013). Modern Engineering Mathematics. 7th Edition. Palgrave MacMilan.

4. E. Kreyszig (2010). Advance Engineering Mathematics. 9th Edition. John Wiley and Son.

First Order ODE

• Introduction to ODEs• Describe the concept of ODEs• Apply an ODEs in real life application

Basic Concepts• In solving engineering problem, the problem could be formulated as a

mathematical expression in terms of variables, functions, equations and so forth.

• Such an expression is known as a mathematical model of the given problem.

• Since many physical concepts, such as velocity and acceleration are derivatives, a model is very often an equation containing derivatives of an unknown function.

• Such a model is called a differential equation.

• An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation.

What is ODE?• If an equation contains only ordinary derivatives

of one or more dependent variables with respect to a single independent variable; it is said to be an ordinary differential equation (ODE).

• An ordinary differential equation (ODE) is a relationship between an independent variable, x, a dependent variable y, and one or more derivatives of y with respect to x. The equation may also contain y itself, known functions of x (or t), and constants. Examples:

A DE can contain more than one dependent variable

What is ODE?

• ODE is different with partial differential equations (PDE) which involve partial derivatives of an unknown function of two or more variables. For example:

Order of an ODE

• An ODE is said to be of order n if the nth derivative of the unknown function y is the highest derivative of y in the equation.

second order first order

Order of an ODE

• Example:

Linearity of ODE• An nth-order ordinary differential equation is said to be linear if the

function is linear in y, y’, …….y(n). This means that an nth-order ODE is linear when

• The characteristic two properties of a linear ODE are:

a) The dependent variable y and all its derivatives y’, y’’,…y(n) are of the first degree, that is the power of each term involving y is 1.

b) The coefficients a0, a1,…an of y, y’,…y(n) depend at most on the independent variable x.

Linearity of ODE

• Example of linear first-, second-, and third-order ODE:

Linearity of ODE

• A nonlinear ODE is simply one that is not linear.

• Nonlinear functions of the dependent variable or its derivatives, such as or cannot appear in a linear equation.

Linearity of ODE• Example

Nonlinear term: coefficient depends on y

Nonlinear term: nonlinear function of y

Nonlinear term: power not 1

Concept of solution• A solution containing a number of independent arbitrary constants equal to the

order of the differential equation is called the general solution of the equation.

• Any function y(x) with N arbitrary constants in it be a general solution of a N th order ODE in y = y(x) if the function satisfies the ODE.

• Example

Where C and D are arbitrary constants.

is the general solution of the ODE

Concept of solution

• When specific values are given to at least one of these arbitrary constants, the solution is called a particular solution.

• Example:

are examples of particular solutions of the ODE

Concept of solution• A solution of an ODE is exact if it (the solution) can be expressed in terms

of elementary functions.

• A function is elementary if its value can be calculated using ordinary scientific hand calculator.

• Thus the general solution of the ODE is exact.

• Some ODEs may not have exact solution, example:

Explicit and Implicit Solutions

• A solution in which the dependent variable is expressed solely in terms of the independent variable and constants is said to be explicit solution.

is the explicit solution of

is the explicit solution of

Explicit and Implicit Solutions• A relation is said to be an implicit solution of an ordinary differential

equation on an interval I, provided that there exists at least one function Ø that satisfies the relation as well as the differential equation on I.

is an implicit solution for on an open interval (-5,5).

Homogeneity of ODE

• Linear ODE is said to be homogeneous if the function for is given by for all That is:

Homogeneity of ODE

• If the linear ODE is such that the function can have non-zero value for at least some values of ; then the ODE is said to be inhomogeneous (or nonhomogeneous).

Real life applications

Example of applications• Falling object under the influence of gravity

• Assuming air resistance is so small and negligible; then the acceleration of the object is the same as acceleration of gravity:

(t is time)

• By integration, where is the initial velocity with which motion started (e.g. )

• Integrating once more to get the distance travelled: where is the distance from 0 at the beginning