MATH105: ENGINEERING MATHEMATICS II

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Course Prerequisite(s)

About Course

This class is to experience an introduction to the rich, complex,
and powerful subject of Ordinary Differential Equations (ODEs). Specifically:
(1) Develop a working familiarity with linear algebra to the extent we need it
for the differential equations we shall consider. Linear algebra serves us as a very robust backend for handling all higher-dimensional linear issues which will arise. (2)Most “real life” systems that are described mathematically, be they physical, biological, financial or economic, are described by means of differential equations. Our ability to predict the way in which these systems evolve or behave is determined by our ability to model these systems and find solutions of the equations explicitly or approximately. Every application and differential equation presents its own challenges, but there are various classes of differential equations, and for some of these there are established approaches and methods for solving them.

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What Will You Learn?

  • Upon successful completion of this course, a student will be able to:
  • Create and analyze mathematical models using ordinary differential equations;1.
  • Identify the type of a given differential equation and select and apply the appropriate analytical technique
  • for finding the solution of first order and selected higher order ordinary differential equations;
  • 2.Apply the existence and uniqueness theorems for ordinary differential equations;3.
  • Explain the meaning of existence and uniqueness.4.
  • Determine the Laplace Transform and inverse Laplace Transform of functions; and5.
  • Solve Linear Systems of ordinary differential equations.

Course Content

Systems of Equations
In this section we’ll introduce most of the basic topics that we’ll need in order to solve systems of equations including augmented matrices and row operations

Solving Systems of Equations
Here we will look at the Gaussian Elimination and Gauss-Jordan Method of solving systems of equations.

Matrix Arithmetic & Operations
In this section we’ll take a look at matrix addition, subtraction and multiplication. We’ll also take a quick look at the transpose and trace of a matrix

Inverse Matrices and Elementary Matrices
Here we’ll define the inverse and take a look at some of its properties. We’ll also introduce the idea of Elementary Matrices.

Finding Inverse Matrices
In this section we’ll develop a method for finding inverse matrices.

Special Matrices
We will introduce Diagonal, Triangular and Symmetric matrices in this section.

LU-Decompositions
In this section we’ll introduce the LU-Decomposition a way of “factoring” certain kinds of matrices.

Limit and Continuity of Multivariable functions
The topic that we will be examining in this chapter is that of Limits. This is the first of three major topics that we will be covering in this course. While we will be spending the least amount of time on limits in comparison to the other two topics limits are very important in the study of Calculus. We will be seeing limits in a variety of places once we move out of this chapter. In particular we will see that limits are part of the formal definition of the other two major topics.

The Derivatives of multivariable function
In this chapter we will start looking at the next major topic in a calculus class, derivatives. This chapter is devoted almost exclusively to finding derivatives. We will be looking at one application of them in this chapter. We will be leaving most of the applications of derivatives to the next chapter.

Applications Of Derivatives
In the previous chapter we focused almost exclusively on the computation of derivatives. In this chapter will focus on applications of derivatives. It is important to always remember that we didn’t spend a whole chapter talking about computing derivatives just to be talking about them. There are many very important applications to derivatives.

Integrals of multivariable functions and applications
In the past two chapters we’ve been given a function, , and asking what the derivative of this function was. Starting with this section we are now going to turn things around. We now want to ask what function we differentiated to get the function

First-order ordinary differential equations
Solutions of ordinary differential equations

Applications of first order differential equations such as circuits, mixture problems, population modeling,

Second order and higher order linear differential equation
Applications of first order differential equations such as circuits, mixture problems, population modeling,

Applications of higher order differential equations such as the harmonic oscillator and circuits

LaplaceTransforms

Series Solutions

Systems of Ordinary differential equations