ODEs and Linear Algebra Assignment

Read these instructions carefully

• The presentation of your answers matters a lot – you must explain what you are doing and you must use proper mathematical notation (as used in texts, notes etc). Just writing an answer without working is not enough.

Guidelines for submitted work

• This is an academic assignment so it must be referenced. Include a Reference list at the end of your assignment. You must cite all the different authors from different sources (books, journals, electronic). Please do not use Wikipedia or other such non-refereed sources.

• Start each new question on a new page.

• Conduct a spell check yourself and ensure you have a critical friend read and comment on your English usage, grammar, punctuation and other technical issues.

• Please use Microsoft Word Equation to express any mathematical formulas needed, and Microsoft Word to write the assignment or similar software.

• You may use graphical or drawing software to show your graphs.

• Please use size 12 font for any written work.

• Leave a wide margin on the left for feedback comments from your teachers.

What is plagiarism

Plagiarism is when you copy someone else’s answers. Even if you make slight changes in symbols it is still plagiarism. Plagiarism is cheating and is wrong. If it is detected all the people whose answers are extremely similar will get zero marks for the questions involved.

It is a good idea to discuss problems with other people. It is often helpful to work in study groups, but you must write up your answers by yourself and your examples must be unique. Any similarities identified with TURNITIN will be investigated and penalties will be applied.

ODEs and Linear Algebra Assignment

You have been hired to write an introduction to the section on Differential Equations and Linear Algebra of this course.

NO CALCULATORS TO BE USED, THAT IS, USE NUMBERS THAT ARE SMALL AND EASY TO DO CALCULATIONS WITH MANUALLY.

Your examples must not be ones included in the material for this course; you are expected to find, or make up, your own examples. You must also include references for all questions and a proper referencing style must be used such as AP A.

Question 1

• In your own words explain what are differential equations and why they are important to engineering. At least one paragraph fully referenced is required.

• Write a differential equation for a problem like this: A person is trying to fill a bathtub with water. Water is flowing into the bathtub from the tap at a constant rate of k litres/sec. However, there is a hole in the bottom of the bathtub and water is flowing out of the bathtub at a rate proportional to the square of the volume of water present in the bathtub. If V(t) is the volume of water (in litres) present in the bathtub at time t (in seconds) and the bathtub initially contains V(0) litres of water, write down, but do not solve, the differential equation for V(t) along with its initial condition.

Question 2

Explain what is a logistic differential equation. Use your own example to show how it is solved and what does it represent.

Question 3

Use your own examples to show how to solve one first order and one second order, linear, inhomogeneous with constant coefficients, ordinary differential equation.

Question 4

Use your own examples to explain to a first year engineering student what are eigenvalues and eigenvectors in 2D only, and explain how and why they are important in engineering.

THE END

Performance Objectives:

Know:

Why are ODEs needed

How to solve ODEs analytically

How to use eigenvalues and eigenvectors