Homework 4 Solving Systems By Elimination
Now, with the basic terms out of the way, let's get to solving! Suppose we have two dependent variables, x and y, representing the population of krills and yaks. Imagine the rate of change of yaks, dx/dt, depending negatively on the number of krills. Similarly, in this hypothetical world, the rate of change of krills, dy/dt, depends negatively on the number of yaks. Thus, our hypothetical coupled system of linear differential equations is:
homework 4 solving systems by elimination
Two unknowns and two equations suggests the elimination method from algebra. As we'll see, writing dx/dt as Dx looks like D is multiplying x. The D, however, is an operator on x where the operation is differentiation. Actually, even multiplication, like the 4 times x in 4x, is a 4 operating on the x.
This is analogous to the systems of equations encountered in algebra. To solve a system of differential equations, borrow algebra's elimination method. Derivatives like dx/dt are written as Dx and the operator D is treated like a multiplying constant.
Using elimination, the system of differential equations is reduced to one differential equation in one variable. Standard methods are used to solve this differential equation. The resulting solution contains unknown constants determined using initial conditions, which are the variables and its first derivative values at time t = 0. The process of eliminating variables, solving a differential equation, and finding values for the constants, is repeated until all the dependent variables are solved.
When dealing with a system of linear equations there are two methods to algebraically solve the question. One is substitution and the other is elimination which is meant to be a shortcut. Both methods will bring you to the same solution but with more practice, you will recognize patterns and see which method would work best when given a system. The best way to show how to solve these kinds of questions are by providing an example to work on.
There you go, both methods get you the same answer whenever asked to solve a linear system. Notice that when x or y has no coefficient, then substitution would be faster. If the x or y of both lines are the same then elimination would be faster.
1/27: In your homework, you will be applying row operations tomatrices, and looking at matrices which may or may not be in rowreduced echelon form (r.r.e.f.). Check out this online application which gives you the steps forturning a matrix into r.r.e.f. Try some small examples to see whatr.r.e.f. looks like. Can you think of a methodical procedure to turnany 2x3 matrix into r.r.e.f. with elementary row operations?
1/30: In lecture, we discussed a procedure which turns any matrix intoa matrix in row reduced echeclon form called Gaussian elimination. Itis interesting to note that, even in the development of Europeanmathematics, Carl Friedrich Gauss was not the first mathematician towrite down a general procedure for row reducing a matrix. You can readmore about the history and development of solving systems of linearequations with elimination methods in this article by Joseph Grcar.
2/3: Due to inclement weather, class was cancelled. Please have yourhomework ready to hand in on Thursday. Please ensure that all pages ofyour assignment are stapled together, or similarly securely attachedtogether. We will resume on Thursday 2/6 with section 1.6.
2/13: Due to inclement weather, class was cancelled. The homeworkassignment is now due on Monday 2/17. The first midterm is stillscheduled for 2/27 during our usual class time. The topic of LU decomposition has been dropped. It is a very useful thing to knowif you are ever doing computer calculations to solve systems of linearequations. I suggest you look it over if you have time.
The suggested homework problems for each section can be found here (.pdf). You are expected to work through, andturn in all of the problems in regular weight font (non-bold). Aselection of these problems will be graded on content. I will make theselected problems known to you in class. The homework sets are to beturned in every Monday, and consist of the problems from the previousweek's lecture topics. No late assignments will be accepted. Instead,your two lowest homework grades will be dropped. Homework accounts for15% of your overall grade.
Homework will be assigned regularly, and will primarilyconsist of written exercises similar to those in the book.Assignments will be discussed in class and posted on thehomework page. They are due atthe beginning of class on the indicated due date; no lateassignments will be accepted. Solution keys for eachassignment will be posted for help in studying for exams.
We will have 3 in-class exams, the dates of which are posted on the course web page. The exam questions will be based off of the homeworks and projects. These exams will be non-cumulative, and will be open-book/open-note. There will be no final exam.
Examples of honor code violations include:Submitting a computer project which includes a program, or even part of a program, written by someone else (other than those provided by the instructor). This includes programs written by students from previous semesters, and programs downloaded from the internet.Submitting computer outputs (numerical results or plots) produced by someone else's program.
Submitting homework answers that are copied from another student's work.
Supplying your own work for another student to copy.
A generally applicable rule of thumb in this course is: you are encouraged to talk about homework/project strategy all you want, but you should never look at another student's work to be submitted for a grade.